text:
(* Author: Alexander Maletzky *)

section \<open>Degree Sections of Power-Products\<close>

theory Degree_Section
  imports Polynomials.MPoly_PM
begin

definition deg_sect :: "'x set \<Rightarrow> nat \<Rightarrow> ('x::countable \<Rightarrow>\<^sub>0 nat) set"
  where "deg_sect X d = .[X] \<inter> {t. deg_pm t = d}"

definition deg_le_sect :: "'x set \<Rightarrow> nat \<Rightarrow> ('x::countable \<Rightarrow>\<^sub>0 nat) set"
  where "deg_le_sect X d = (\<Union>d0\<le>d. deg_sect X d0)"

lemma deg_sectI: "t \<in> .[X] \<Longrightarrow> deg_pm t = d \<Longrightarrow> t \<in> deg_sect X d"
  by (simp add: deg_sect_def)

lemma deg_sectD:
  assumes "t \<in> deg_sect X d"
  shows "t \<in> .[X]" and "deg_pm t = d"
  using assms by (simp_all add: deg_sect_def)

lemma deg_le_sect_alt: "deg_le_sect X d = .[X] \<inter> {t. deg_pm t \<le> d}"
  by (auto simp: deg_le_sect_def deg_sect_def)

lemma deg_le_sectI: "t \<in> .[X] \<Longrightarrow> deg_pm t \<le> d \<Longrightarrow> t \<in> deg_le_sect X d"
  by (simp add: deg_le_sect_alt)

lemma deg_le_sectD:
  assumes "t \<in> deg_le_sect X d"
  shows "t \<in> .[X]" and "deg_pm t \<le> d"
  using assms by (simp_all add: deg_le_sect_alt)

lemma deg_sect_zero [simp]: "deg_sect X 0 = {0}"
  by (auto simp: deg_sect_def zero_in_PPs)

lemma deg_sect_empty: "deg_sect {} d = (if d = 0 then {0} else {})"
  by (auto simp: deg_sect_def)

lemma deg_sect_singleton [simp]: "deg_sect {x} d = {Poly_Mapping.single x d}"
  by (auto simp: deg_sect_def deg_pm_single PPs_singleton)

lemma deg_le_sect_zero [simp]: "deg_le_sect X 0 = {0}"
  by (auto simp: deg_le_sect_def)

lemma deg_le_sect_empty [simp]: "deg_le_sect {} d = {0}"
  by (auto simp: deg_le_sect_alt varnum_eq_zero_iff)

lemma deg_le_sect_singleton: "deg_le_sect {x} d = Poly_Mapping.single x ` {..d}"
  by (auto simp: deg_le_sect_alt deg_pm_single PPs_singleton)

lemma deg_sect_mono: "X \<subseteq> Y \<Longrightarrow> deg_sect X d \<subseteq> deg_sect Y d"
  by (auto simp: deg_sect_def dest: PPs_mono)

lemma deg_le_sect_mono_1: "X \<subseteq> Y \<Longrightarrow> deg_le_sect X d \<subseteq> deg_le_sect Y d"
  by (auto simp: deg_le_sect_alt dest: PPs_mono)

lemma deg_le_sect_mono_2: "d1 \<le> d2 \<Longrightarrow> deg_le_sect X d1 \<subseteq> deg_le_sect X d2"
  by (auto simp: deg_le_sect_alt)

lemma zero_in_deg_le_sect: "0 \<in> deg_le_sect n d"
  by (simp add: deg_le_sect_alt zero_in_PPs)

lemma deg_sect_disjoint: "d1 \<noteq> d2 \<Longrightarrow> deg_sect X d1 \<inter> deg_sect Y d2 = {}"
  by (auto simp: deg_sect_def)

lemma deg_le_sect_deg_sect_disjoint: "d1 < d2 \<Longrightarrow> deg_le_sect Y d1 \<inter> deg_sect X d2 = {}"
  by (auto simp: deg_sect_def deg_le_sect_alt)

lemma deg_sect_Suc:
  "deg_sect X (Suc d) = (\<Union>x\<in>X. (+) (Poly_Mapping.single x 1) ` deg_sect X d)" (is "?A = ?B")
proof (rule set_eqI)
  fix t
  show "t \<in> ?A \<longleftrightarrow> t \<in> ?B"
  proof
    assume "t \<in> ?A"
    hence "t \<in> .[X]" and "deg_pm t = Suc d" by (rule deg_sectD)+
    from this(2) have "keys t \<noteq> {}" by auto
    then obtain x where "x \<in> keys t" by blast
    hence "1 \<le> lookup t x" by (simp add: in_keys_iff)
    from \<open>t \<in> .[X]\<close> have "keys t \<subseteq> X" by (rule PPsD)
    with \<open>x \<in> keys t\<close> have "x \<in> X" ..
    let ?s = "Poly_Mapping.single x (1::nat)"
    from \<open>1 \<le> lookup t x\<close> have "?s adds t"
      by (auto simp: lookup_single when_def intro!: adds_poly_mappingI le_funI)
    hence t: "?s + (t - ?s) = t" by (metis add.commute adds_minus)
    have "t - ?s \<in> deg_sect X d"
    proof (rule deg_sectI)
      from \<open>t \<in> .[X]\<close> show "t - ?s \<in> .[X]" by (rule PPs_closed_minus)
    next
      from deg_pm_plus[of ?s "t - ?s"] have "deg_pm t = Suc (deg_pm (t - ?s))"
        by (simp only: t deg_pm_single)
      thus "deg_pm (t - ?s) = d" by (simp add: \<open>deg_pm t = Suc d\<close>)
    qed
    hence "?s + (t - ?s) \<in> (+) ?s ` deg_sect X d" by (rule imageI)
    hence "t \<in> (+) ?s ` deg_sect X d" by (simp only: t)
    with \<open>x \<in> X\<close> show "t \<in> ?B" ..
  next
    assume "t \<in> ?B"
    then obtain x where "x \<in> X" and "t \<in> (+) (Poly_Mapping.single x 1) ` deg_sect X d" ..
    from this(2) obtain s where s: "s \<in> deg_sect X d"
      and t: "t = Poly_Mapping.single x 1 + s" (is "t = ?s + s") ..
    show "t \<in> ?A"
    proof (rule deg_sectI)
      from \<open>x \<in> X\<close> have "?s \<in> .[X]" by (rule PPs_closed_single)
      moreover from s have "s \<in> .[X]" by (rule deg_sectD)
      ultimately show "t \<in> .[X]" unfolding t by (rule PPs_closed_plus)
    next
      from s have "deg_pm s = d" by (rule deg_sectD)
      thus "deg_pm t = Suc d" by (simp add: t deg_pm_single deg_pm_plus)
    qed
  qed
qed

lemma deg_sect_insert:
  "deg_sect (insert x X) d = (\<Union>d0\<le>d. (+) (Poly_Mapping.single x (d - d0)) ` deg_sect X d0)"
    (is "?A = ?B")
proof (rule set_eqI)
  fix t
  show "t \<in> ?A \<longleftrightarrow> t \<in> ?B"
  proof
    assume "t \<in> ?A"
    hence "t \<in> .[insert x X]" and "deg_pm t = d" by (rule deg_sectD)+
    from this(1) obtain e tx where "tx \<in> .[X]" and t: "t = Poly_Mapping.single x e + tx"
      by (rule PPs_insertE)
    have "e + deg_pm tx = deg_pm t" by (simp add: t deg_pm_plus deg_pm_single)
    hence "e + deg_pm tx = d" by (simp only: \<open>deg_pm t = d\<close>)
    hence "deg_pm tx \<in> {..d}" and e: "e = d - deg_pm tx" by simp_all
    from \<open>tx \<in> .[X]\<close> refl have "tx \<in> deg_sect X (deg_pm tx)" by (rule deg_sectI)
    hence "t \<in> (+) (Poly_Mapping.single x (d - deg_pm tx)) ` deg_sect X (deg_pm tx)"
      unfolding t e by (rule imageI)
    with \<open>deg_pm tx \<in> {..d}\<close> show "t \<in> ?B" ..
  next
    assume "t \<in> ?B"
    then obtain d0 where "d0 \<in> {..d}" and "t \<in> (+) (Poly_Mapping.single x (d - d0)) ` deg_sect X d0" ..
    from this(2) obtain s where s: "s \<in> deg_sect X d0"
      and t: "t = Poly_Mapping.single x (d - d0) + s" (is "t = ?s + s") ..
    show "t \<in> ?A"
    proof (rule deg_sectI)
      have "?s \<in> .[insert x X]" by (rule PPs_closed_single, simp)
      from s have "s \<in> .[X]" by (rule deg_sectD)
      also have "... \<subseteq> .[insert x X]" by (rule PPs_mono, blast)
      finally have "s \<in> .[insert x X]" .
      with \<open>?s \<in> .[insert x X]\<close> show "t \<in> .[insert x X]" unfolding t by (rule PPs_closed_plus)
    next
      from s have "deg_pm s = d0" by (rule deg_sectD)
      moreover from \<open>d0 \<in> {..d}\<close> have "d0 \<le> d" by simp
      ultimately show "deg_pm t = d" by (simp add: t deg_pm_single deg_pm_plus)
    qed
  qed
qed

lemma deg_le_sect_Suc: "deg_le_sect X (Suc d) = deg_le_sect X d \<union> deg_sect X (Suc d)"
  by (simp add: deg_le_sect_def atMost_Suc Un_commute)

lemma deg_le_sect_Suc_2:
  "deg_le_sect X (Suc d) = insert 0 (\<Union>x\<in>X. (+) (Poly_Mapping.single x 1) ` deg_le_sect X d)"
    (is "?A = ?B")
proof -
  have eq1: "{Suc 0..Suc d} = Suc ` {..d}" by (simp add: image_Suc_atMost)
  have "insert 0 {1..Suc d} = {..Suc d}" by fastforce
  hence "?A = (\<Union>d0\<in>insert 0 {1..Suc d}. deg_sect X d0)" by (simp add: deg_le_sect_def)
  also have "... = insert 0 (\<Union>d0\<le>d. deg_sect X (Suc d0))" by (simp add: eq1)
  also have "... = insert 0 (\<Union>d0\<le>d. (\<Union>x\<in>X. (+) (Poly_Mapping.single x 1) ` deg_sect X d0))"
    by (simp only: deg_sect_Suc)
  also have "... = insert 0 (\<Union>x\<in>X. (+) (Poly_Mapping.single x 1) ` (\<Union>d0\<le>d. deg_sect X d0))"
    by fastforce
  also have "... = ?B" by (simp only: deg_le_sect_def)
  finally show ?thesis .
qed

lemma finite_deg_sect:
  assumes "finite X"
  shows "finite ((deg_sect X d)::('x::countable \<Rightarrow>\<^sub>0 nat) set)"
proof (induct d)
  case 0
  show ?case by simp
next
  case (Suc d)
  with assms show ?case by (simp add: deg_sect_Suc)
qed

corollary finite_deg_le_sect: "finite X \<Longrightarrow> finite ((deg_le_sect X d)::('x::countable \<Rightarrow>\<^sub>0 nat) set)"
  by (simp add: deg_le_sect_def finite_deg_sect)

lemma keys_subset_deg_le_sectI:
  assumes "p \<in> P[X]" and "poly_deg p \<le> d"
  shows "keys p \<subseteq> deg_le_sect X d"
proof
  fix t
  assume "t \<in> keys p"
  also from assms(1) have "... \<subseteq> .[X]" by (rule PolysD)
  finally have "t \<in> .[X]" .
  from \<open>t \<in> keys p\<close> have "deg_pm t \<le> poly_deg p" by (rule poly_deg_max_keys)
  from this assms(2) have "deg_pm t \<le> d" by (rule le_trans)
  with \<open>t \<in> .[X]\<close> show "t \<in> deg_le_sect X d" by (rule deg_le_sectI)
qed

lemma binomial_symmetric_plus: "(n + k) choose n = (n + k) choose k"
  by (metis add_diff_cancel_left' binomial_symmetric le_add1)

lemma card_deg_sect:
  assumes "finite X" and "X \<noteq> {}"
  shows "card (deg_sect X d) = (d + (card X - 1)) choose (card X - 1)"
  using assms
proof (induct X arbitrary: d)
  case empty
  thus ?case by simp
next
  case (insert x X)
  from insert(1, 2) have eq1: "card (insert x X) = Suc (card X)" by simp
  show ?case
  proof (cases "X = {}")
    case True
    thus ?thesis by simp
  next
    case False
    with insert.hyps(1) have "0 < card X" by (simp add: card_gt_0_iff)
    let ?f = "\<lambda>d0. Poly_Mapping.single x (d - d0)"
    from False have eq2: "card (deg_sect X d0) = d0 + (card X - 1) choose (card X - 1)" for d0
      by (rule insert.hyps)
    have "finite {..d}" by simp
    moreover from insert.hyps(1) have "\<forall>d0\<in>{..d}. finite ((+) (?f d0) ` deg_sect X d0)"
      by (simp add: finite_deg_sect)
    moreover have "\<forall>d0\<in>{..d}. \<forall>d1\<in>{..d}. d0 \<noteq> d1 \<longrightarrow>
                          ((+) (?f d0) ` deg_sect X d0) \<inter> ((+) (?f d1) ` deg_sect X d1) = {}"
    proof (intro ballI impI, rule ccontr)
      fix d1 d2 :: nat
      assume "d1 \<noteq> d2"
      assume "((+) (?f d1) ` deg_sect X d1) \<inter> ((+) (?f d2) ` deg_sect X d2) \<noteq> {}"
      then obtain t where "t \<in> ((+) (?f d1) ` deg_sect X d1) \<inter> ((+) (?f d2) ` deg_sect X d2)"
        by blast
      hence t1: "t \<in> (+) (?f d1) ` deg_sect X d1" and t2: "t \<in> (+) (?f d2) ` deg_sect X d2" by simp_all
      from t1 obtain s1 where "s1 \<in> deg_sect X d1" and s1: "t = ?f d1 + s1" ..
      from this(1) have "s1 \<in> .[X]" by (rule deg_sectD)
      hence "keys s1 \<subseteq> X" by (rule PPsD)
      with insert.hyps(2) have eq3: "lookup s1 x = 0" by (auto simp: in_keys_iff)
      from t2 obtain s2 where "s2 \<in> deg_sect X d2" and s2: "t = ?f d2 + s2" ..
      from this(1) have "s2 \<in> .[X]" by (rule deg_sectD)
      hence "keys s2 \<subseteq> X" by (rule PPsD)
      with insert.hyps(2) have eq4: "lookup s2 x = 0" by (auto simp: in_keys_iff)
      from s2 have "lookup (?f d1 + s1) x = lookup (?f d2 + s2) x" by (simp only: s1)
      hence "d - d1 = d - d2" by (simp add: lookup_add eq3 eq4)
      moreover assume "d1 \<in> {..d}" and "d2 \<in> {..d}"
      ultimately have "d1 = d2" by simp
      with \<open>d1 \<noteq> d2\<close> show False ..
    qed
    ultimately have "card (deg_sect (insert x X) d) =
                       (\<Sum>d0\<le>d. card ((+) (monomial (d - d0) x) ` deg_sect X d0))"
      unfolding deg_sect_insert by (rule card_UN_disjoint)
    also from refl have "... = (\<Sum>d0\<le>d. card (deg_sect X d0))"
    proof (rule sum.cong)
      fix d0
      have "inj_on ((+) (monomial (d - d0) x)) (deg_sect X d0)" by (rule, rule add_left_imp_eq)
      thus "card ((+) (monomial (d - d0) x) ` deg_sect X d0) = card (deg_sect X d0)"
        by (rule card_image)
    qed
    also have "... = (\<Sum>d0\<le>d. (card X - 1) + d0 choose (card X - 1))" by (simp only: eq2 add.commute)
    also have "... = (\<Sum>d0\<le>d. (card X - 1) + d0 choose d0)" by (simp only: binomial_symmetric_plus)
    also have "... = Suc ((card X - 1) + d) choose d" by (rule sum_choose_lower)
    also from \<open>0 < card X\<close> have "... = d + (card (insert x X) - 1) choose d"
      by (simp add: eq1 add.commute)
    also have "... = d + (card (insert x X) - 1) choose (card (insert x X) - 1)"
      by (fact binomial_symmetric_plus)
    finally show ?thesis .
  qed
qed

corollary card_deg_sect_Suc:
  assumes "finite X"
  shows "card (deg_sect X (Suc d)) = (d + card X) choose (Suc d)"
proof (cases "X = {}")
  case True
  thus ?thesis by (simp add: deg_sect_empty)
next
  case False
  with assms have "0 < card X" by (simp add: card_gt_0_iff)
  from assms False have "card (deg_sect X (Suc d)) = (Suc d + (card X - 1)) choose (card X - 1)"
    by (rule card_deg_sect)
  also have "... = (Suc d + (card X - 1)) choose (Suc d)" by (rule sym, rule binomial_symmetric_plus)
  also from \<open>0 < card X\<close> have "... = (d + card X) choose (Suc d)" by simp
  finally show ?thesis .
qed

corollary card_deg_le_sect:
  assumes "finite X"
  shows "card (deg_le_sect X d) = (d + card X) choose card X"
proof (induct d)
  case 0
  show ?case by simp
next
  case (Suc d)
  from assms have "finite (deg_le_sect X d)" by (rule finite_deg_le_sect)
  moreover from assms have "finite (deg_sect X (Suc d))" by (rule finite_deg_sect)
  moreover from lessI have "deg_le_sect X d \<inter> deg_sect X (Suc d) = {}"
    by (rule deg_le_sect_deg_sect_disjoint)
  ultimately have "card (deg_le_sect X (Suc d)) = card (deg_le_sect X d) + card (deg_sect X (Suc d))"
    unfolding deg_le_sect_Suc by (rule card_Un_disjoint)
  also from assms have "... = (Suc d + card X) choose Suc d"
    by (simp add: Suc.hyps card_deg_sect_Suc binomial_symmetric_plus[of d])
  also have "... = (Suc d + card X) choose card X" by (rule binomial_symmetric_plus)
  finally show ?case .
qed

end (* theory *)

subset_name: curated


TITLE: How to present a paper?
QUESTION [7 upvotes]: I am going to present several papers to an audience. I have read through all the papers and have a clear idea about them.
But this is the first time I have ever presented papers, and I am guessing that many of you have done this many times. So I am sure I can get a lot of good advice from you on how to do this.
I have about 80min to talk about 2-3 papers. They are about operator algebras and my audience are experts who have been working in this field for at least 20 years.
Any advice would help!
Thanks!

REPLY [5 votes]: This sounds like a presentation for an oral examination.  I'm going to give you some tips I found helpful in preparing my oral exam presentation, which was over one paper and several supporting papers.  (I am assuming that your goal is to present papers, not proofs.  I don't mean to say, don't include proofs when you're presenting several papers, but that if you focus on the details of any proofs, you run the risk of losing your audience's interest in a thicket of technicalities.  rschwieb has a very good comment below.)
Rehearse the talk beforehand.  Make sure that you don't run over time.
You have probably spent weeks reading these papers.  You need to distill those weeks down into 80 minutes of clearly explained ideas.  If I were giving the presentation, I would give the big-picture logical structure of the papers.  I wouldn't present any details unless you are asked.
I'd try as best I could to make that big picture logical structure a story.  Mathematicians are just human, after all, and we follow stories better than logic.  What motivates these papers?  What common themes tie them together?  The motivation is the problem you have to solve, the themes are the characters, the plot is how the themes come together to solve the problem.
I think that finding common themes and turning them into a story is the most important part of this exercise.  By doing this, you'll get to show off how discerningly you know the math (since you have to know it well in order to distill it down to common themes and then focus on those).  You'll also have the pleasure of stepping back and looking at the big picture as you prepare, instead of losing yourself in yucky details.  Finally, you'll also be able to tell a (hopefully semi-exciting) story instead of just reciting "theorem ... proof ... lemma ... proof ... theorem ... proof ..." while your audience dozes off.
One other tip.  Before starting, I'd outline the talk narrative and every five or ten minutes, as the flow of the material allows, remind the audience where we are in the narrative.  This helps put the story in perspective, is a temporary check for anybody tuning out, and (frankly) reminds the audience that the talk isn't going to last forever.  It also helps hammer the chief ideas home.  In any speech, you do three things: tell them what you're going to tell them, tell them, and then tell them what you told them.


text:
\begin{document}

\title{A Minimal Model Program for $\Qbb$-Gorenstein varieties}

\author{Boris Pasquier\protect\footnote{Boris PASQUIER,
E-mail: boris.pasquier@univ-montp2.fr}}

\maketitle
\begin{abstract}
The main results of this paper are already known \cite{Shokurov}. Moreover, the non-$\mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties \cite{Fujino1}, for klt pairs \cite{Fujino2} and more generally for log-canonical pairs \cite{Fujino3}.

Here we rewrite the proofs of some of these results, by following the proofs given in \cite{KMM} of the same results in $\mathbb{Q}$-factorial MMP. And, in the family of $\mathbb{Q}$-Gorenstein spherical varieties, we answer positively to the questions of existence of flips and of finiteness of sequences of flips.

I apologize for the first version of this paper, which I wrote without knowing that these results already exist.
\end{abstract}

\textbf{Mathematics Subject Classification.} 14E30 14M25 14M17
\\

\textbf{Keywords.} Minimal Model Program.\\

\section{Introduction}

We only consider normal algebraic varieties over $\Cbb$.

In this paper we prove some already well-known results on $\Qbb$-Gorenstein MMP (also called non-$\mathbb{Q}$-factorial MMP).
The motivation comes from \cite{MMPhoro}, where the MMP for horospherical varieties is reduced to the study of a family of moment polytopes, $\Qbb$-Gorenstein MMP seams to be more natural than $\Qbb$-factorial MMP.

Then we answer positively to the questions of existence of flips and of finiteness of sequences of flips for spherical varieties.

Before to describe explicitly the results, we recall some basic definitions and notations.

\begin{defi}
\begin{itemize}
\item A normal variety $X$ is $\Qbb$-Gorenstein if its canonical divisor $K_X$ is $\Qbb$-Cartier (ie if there exists a multiple of $K_X$ that is Cartier).
\item A normal $\Qbb$-Gorenstein variety $X$ has terminal singularities if there exists a desingularization $\sigma:\,V\longrightarrow X$ of $X$ such that $$K_V=\sigma^*K_X+\sum_{E\,{\rm prime}\,{\rm divisor}\,{\rm of }\,V}a_EE,$$
with $a_E>0$ for any exceptional prime divisor $E$ of $\sigma$ (and $a_E=0$ otherwise).
\item A contraction (ie a projective morphism $\phi:\,X\longrightarrow Y$ such that $\phi_*\mathcal{O}_X=\mathcal{O}_Y$) is a small contraction if its exceptional locus is at least of codimension~2.
\end{itemize}
\end{defi}

Now, fix a normal $\Qbb$-Gorenstein projective variety $X$ with terminal singularities.
We denote by $NE(X)$ the nef cone of curves of $X$ and by $NE(X)_{K_X<0}$ (resp. $NE(X)_{K_X>0}$) the intersection of the nef cone with the open half-space of curves negative (resp. positive) along the divisor $K_X$. Then, by the Cone Theorem, $NE(X)=NE(X)_{K_X>0}+\sum_{i\in I}R_i$, where the set $\{R_i\,\mid\,i\in I\}$ is a discrete set of (extremal) rays of $NE(X)_{K_X<0}$. Fix a ray $R$ of $NE(X)_{K_X<0}$. Then, by the Contraction Theorem, there exists a unique contraction $\phi:\,X\longrightarrow Y$, such that, for any curve $C$ of $X$, $\phi(C)$ is a point if and only if the class of $C$ is in $R$.

We recall the definition of flips we use in this paper.
\begin{defi}\label{def:QGorflip}
Let $X$ be a normal $\Qbb$-Gorenstein projective variety, and $\phi:X\longrightarrow Y$ be a birational contraction of a ray of $NE(X)_{K_X<0}$, such that $Y$ is not $\Qbb$-Gorenstein.

A flip of $\phi$ is a small contraction $\phi:X^+\longrightarrow Y$, where
\begin{itemize}
\item $X^+$ is a normal $\Qbb$-Gorenstein projective variety;
\item and for any contracted curve $C$ of $X^+$, $K_{X^+}\cdot C>0$.
\end{itemize}
\end{defi}

In this paper, we prove the following (already known \cite{Shokurov}) result.

\begin{teo}\label{th:main} Let $X$ be a normal $\Qbb$-Gorenstein projective variety with terminal singularities. Let $R$ be a ray of $NE(X)_{K_X<0}$ and denote by $\phi:\,X\longrightarrow Y$ the contraction of $R$. Suppose that $\phi$ is birational.
\begin{itemize}
\item If $Y$ is $\Qbb$-Gorenstein, then $Y$ has terminal singularities and $\phi$ contracts a $\Qbb$-Cartier divisor.
\item If $Y$ is not $\Qbb$-Gorenstein, then there exists a flip of $\phi$ if and only if $\oplus_{l\geq 0}\phi_*\mathcal{O}_X(lmK_X)$ is a finitely generated sheaf of $\mathcal{O}_Y$-algebras. Moreover, if it exists, it is unique and $X^+$ has terminal singularities.
\end{itemize}
\end{teo}

The proof (given in this paper) of Theorem~\ref{th:main} is inspired by the proofs of the same results in the original MMP \cite{KMM}, which are also detailed in \cite{matsuki}.\\

The paper is organized as follows.

In Section~\ref{section1}, we list several probably well-known and general results, which are very useful in the rest of the paper.

Sections~\ref{section2} and~\ref{section3} are devoted to the proof of Theorem~\ref{th:main}.

In Section~\ref{section4}, we illustrate the MMP for $\Qbb$-Gorenstein projective varieties in an example of a 3-dimensional toric variety. In particular, we observe that a flip is not necessarily a contraction of an extremal ray.

In Section~\ref{section5}, we explain how to run a $\Qbb$-Gorenstein log MMP.

And in Section~\ref{section6} we answer positively to the questions of existence of flips and of finiteness of sequences of flips.

\section{General lemmas}\label{section1}

We begin by a very classical result.

\begin{lem}\label{lem:eff}
Let $X$ be a normal variety, and let $D$ be a Cartier divisor of $X$.
Then the following assertions are equivalent.
\begin{enumerate}
\item $D$ is effective.
\item $\mathcal{O}_X\subset \mathcal{O}_X(D)$.
\item $\mathcal{O}_X(-D)\subset \mathcal{O}_X$.
\end{enumerate}
\end{lem}

From Lemma~\ref{lem:eff}, we deduce the following, probably well-known, result.

\begin{lem}\label{lem:pulleff}
Let $\phi:\,X\longrightarrow Y$ be a morphism between two normal varieties $X$ and $Y$.
Let $D$ be an effective $\Qbb$-Cartier divisor of $Y$.

Then $\phi^*D$ is an effective $\Qbb$-Cartier divisor of $X$.
\end{lem}

\begin{proof}
Let $m$ be a positive integer such that $mD$ is Cartier. Since $mD$ is effective, we have $\mathcal{O}_Y\subset \mathcal{O}_Y(mD)$. Then the image of $\phi^*(\mathcal{O}_Y)$ in $\phi^*(\mathcal{O}_Y(mD))$, is the pull-back by $\phi$ of the image of $\mathcal{O}_Y$ in $\mathcal{O}_Y(mD)$. But all these invertible sheaves are subsheaves of the constant sheaf $\mathcal{K}=\Cbb(X)$. Thus, $\phi^*(\mathcal{O}_Y)=\mathcal{O}_X$ is contained in $\phi^*(\mathcal{O}_Y(mD))$. Then $\phi^*(mD)$  is effective and also is $\phi^*(D):=\frac{1}{m}\phi^*(mD)$.
\end{proof}

We know prove the key lemma of the paper.

\begin{lem}\label{lem:key}
Let $\phi:\,X\longrightarrow Y$ be a surjective birational morphism between two normal varieties $X$ and $Y$.
Let $D$ be a Cartier divisor of $X$ and denote by $D_Y$ the Weil divisor $\phi_*D$.

Suppose that the morphism $\phi^*\phi_*\mathcal{O}_X(-D)\longrightarrow\mathcal{O}_X(-D)$ is surjective.

Then the image of $\phi^*\mathcal{O}_Y(D_Y)\longrightarrow\mathcal{K}$ is contained in $\mathcal{O}_X(D)$.

In particular, $\phi_*\mathcal{O}_X(D)=\mathcal{O}_Y(D_Y)$.
\end{lem}

Note that $D_Y$ is not necessarily Cartier, and $\mathcal{O}_Y(D_Y)$ is the (not necessarily invertible) subsheaf of the constant sheaf $\mathcal{K}=\Cbb(X)=\Cbb(Y)$ defined by, for any open set $\mathcal{U}$ of $Y$, $$\mathcal{O}_Y(D_Y)(\mathcal{U})=\{f\in\Cbb(Y)\,\mid\,{\rm div}\,f_{|\mathcal{U}}+D_{Y|\mathcal{U}}\geq 0\}.$$

\begin{proof}
Consider the inclusion of sheaves of $\mathcal{O}_Y$-modules $\phi_*\mathcal{O}_X(-D)\subset\mathcal{O}_Y(-D_Y)$. Tensoring by $\mathcal{O}_Y(D_Y)$, and composing by the natural morphism $\mathcal{O}_Y(D_Y)\otimes_{\mathcal{O}_Y}\mathcal{O}_Y(-D_Y)\longrightarrow\mathcal{O}_Y$, we get the natural morphism of sheaves of $\mathcal{O}_Y$-modules:
$$\mathcal{O}_Y(D_Y)\otimes_{\mathcal{O}_Y}\phi_*\mathcal{O}_X(-D)\longrightarrow\mathcal{O}_Y.$$

The pull-back of this morphism $$\phi^*\mathcal{O}_Y(D_Y)\otimes_{\mathcal{O}_X}\phi^*\phi_*\mathcal{O}_X(-D)\longrightarrow\mathcal{O}_X$$ factors through the following surjective morphism $$\phi^*\mathcal{O}_Y(D_Y)\otimes_{\mathcal{O}_X}\phi^*\phi_*\mathcal{O}_X(-D)\longrightarrow\phi^*\mathcal{O}_Y(D_Y)\otimes_{\mathcal{O}_X}\mathcal{O}_X(-D).$$

Then the image of $\phi^*\mathcal{O}_Y(D_Y)\otimes_{\mathcal{O}_X}\mathcal{O}_X(-D)$ in $\mathcal{K}$ is contained in $\mathcal{O}_X$. We tensor by the invertible sheaf $\mathcal{O}_X(D)$, and we get that $\phi^*\mathcal{O}_Y(D_Y)$ maps to $\mathcal{O}_X(D)$.

To prove the last statement we use that, since $\phi$ is surjective and $\phi_*\mathcal{O}_X=\mathcal{O}_Y$, for any sheaf of $\mathcal{O}_Y$-modules $\mathcal{F}$, we have $\phi_*\phi^*\mathcal{F}=\mathcal{F}$.
Hence, $\mathcal{O}_Y(D_Y)=\phi_*\phi^*\mathcal{O}_Y(D_Y)$ maps to $\phi_*\mathcal{O}_X(D)$. Both are subsheaves of $\mathcal{K}$, so we deduce that this map is an inclusion. The other inclusion is obvious, so that $\phi_*\mathcal{O}_X(D)=\mathcal{O}_Y(D_Y)$.
\end{proof}

From these lemmas, we get a useful corollary.

\begin{cor}\label{cor:keyeff}
Let $\phi:\,X\longrightarrow Y$ be a surjective birational morphism between two normal varieties $X$ and $Y$.
Suppose that there exists a positive integer $m$ such that $mK_X$ and $mK_Y$ are Cartier and such that the morphism $\phi^*\phi_*\mathcal{O}_X(-mK_X)\longrightarrow\mathcal{O}_X(-mK_X)$ is surjective.

Then $K_X-\phi^*K_Y$ is effective.
\end{cor}

\begin{proof} We apply Lemma~\ref{lem:key} to $D=mK_X$ (and $D_Y=mK_Y$).
Since $mK_Y$ is Cartier, then $\phi^*\mathcal{O}_Y(mK_Y)$ is a subsheaf of $\mathcal{K}$ and the map $\phi^*\mathcal{O}_Y(mK_Y)\longrightarrow\mathcal{O}_X(mK_X)$ is an inclusion. We deduce that $\mathcal{O}_X\subset\mathcal{O}_X(mK_X)\otimes_{\mathcal{O}_X}\phi^*\mathcal{O}_Y(-mK_Y)$. And we apply Lemma~\ref{lem:eff} to conclude.
\end{proof}

\section{$\Qbb$-Gorenstein divisorial contractions}\label{section2}

In this section we study the contractions that play the role of divisorial contractions.

\begin{defi}
The contraction $\phi:\,X\longrightarrow Y$ is called a $\Qbb$-Gorenstein divisorial contraction if it is birational and $Y$ is $\Qbb$-Gorenstein.
\end{defi}

The aim of this section is to prove that $Y$ has the same singularities as $X$, and that $\phi$ contracts a (not necessarily irreducible) Cartier divisor.

\begin{teo}\label{th:QGorDiv}\cite{Shokurov}
Let $X$ be a normal $\Qbb$-Gorenstein projective variety with terminal singularities. Let $R$ be a ray of $NE(X)_{K_X<0}$ and denote by $\phi:\,X\longrightarrow Y$ the contraction of $R$.

If $\phi$ is a $\Qbb$-Gorenstein divisorial contraction, then $Y$ is $\Qbb$-Gorenstein with terminal singularities and $\phi$ contracts the (not zero) Cartier divisor $E:=K_X-\phi^*K_Y$.
\end{teo}

\begin{proof}
By hypothesis, $E$ is a $\Qbb$-Cartier divisor of $X$ such that $E\cdot C<0$ for any curve $C$ of class in $R$. In particular $E$ is not zero. Write $E=\sum_{i\in I}a_iE_i$ where for any $i\in I$, where $\{E_i\,\mid\,i\in I\}$ is the set of irreducible exceptional divisors of $\phi$.

Let $\sigma:\,V\longrightarrow X$ be a desingularization of $X$. Denote by $F_j$, with $j\in J$ the irreducible exceptional divisors of $\sigma$, and for any $i\in I$, denote by $G_i$ the strict transform of $E_i$ by $\sigma$. Write $K_V-\sigma^*K_X=\sum_{j\in J}b_jF_j$ and $\sigma^*(K_X-\phi^*K_Y)=\sum_{j\in J} c_jF_j+\sum_{i\in I}a_iG_i.$ since $X$ has terminal singularities, for any $j\in J$, the rational numbers $b_j$ are positive. And, by Lemma~\ref{lem:pulleff} and Corollary~\ref{cor:keyeff}, for any $i\in I$ and any $j\in J$, the rational numbers $a_i$  and $c_j$ are non-negative.

Hence, it is now enough to prove that the rational numbers $a_i$ are positive (or non-zero). Let $i_0\in I$. There exists a curve $C$ in $V$ that is contracted by $\phi\circ\sigma$, contained in $G_{i_0}$ but not in $\rm{Exc}(\sigma)\cup\bigcup_{i\in I,\, i\neq i_0}G_i$. In particular, $\sigma(C)$ is a curve of $X$ that is contracted by $\phi$, for any $j\in J$ we have $F_j\cdot C\geq 0$, and for any $i\in I,\, i\neq i_0$ we have $G_i\cdot C\geq 0$. Then, on the one hand $(K_X-\phi^*K_Y)\cdot\sigma(C)=K_X\cdot\sigma(C)$ so that $\sigma^*(K_X-\phi^*K_Y)\cdot C<0$, and on the other hand $$\sigma^*(K_X-\phi^*K_Y)\cdot C=\sum_{i\in I}a_i(G_i\cdot C)+\sum_{j\in J}c_j(F_j\cdot C)\geq a_{i_0}(G_{i_0}\cdot C).$$ We deduce that $a_{i_0}$ cannot be zero (we necessarily have $a_{i_0}>0$ and $G_{i_0}\cdot C<0$).
\end{proof}

\begin{rem}
In the proof of Theorem~\ref{th:QGorDiv}, we actually prove that $E:=K_X-\phi^*K_Y$ is an exceptional and effective $\Qbb$-Cartier divisor of $X$ such that $E\cdot C<0$ for any curve $C$ of $X$ contracted by $\phi$. Also, $E=\sum_{i\in I}a_iE_i$ where for any $i\in I$, $a_i$ is a positive rational number and where $\{E_i\,\mid\,i\in I\}$ is the set of exceptional divisors of $\phi$.
\end{rem}

\section{Flips}\label{section3}

In this section we interest at the existence of flips (see Definition~\ref{def:QGorflip}).

Note that, if $\phi:X\longrightarrow Y$ is a birational contraction of a ray of $NE(X)_{K_X<0}$ and $m$ is a positive integer such that $mK_X$ is Cartier, then by Lemma~\ref{lem:key} applied to $D=mK_X$ (and $D_Y=mK_Y$), we have
$$\oplus_{l\geq 0}\mathcal{O}_Y(lmK_Y)=\oplus_{l\geq 0}\phi_*\mathcal{O}_X(lmK_X).$$
We denote by $\mathcal{A}$ this sheaf of $\mathcal{O}_Y$-algebras.

Then, we get an analogue result as in the classical MMP.

\begin{teo}\label{th:QGorFlip}\cite{Shokurov}
There exists a flip of $\phi$ if and only if $\mathcal{A}$ is finitely generated as sheaf of $\mathcal{O}_Y$-algebras. In that case, the flip is unique, given by $\phi^+:\,X^+:=\rm{Proj}(\mathcal{A})\longrightarrow Y$.

Moreover, if $X$ has terminal singularities, then $X^+$ has also terminal singularities.
\end{teo}

\begin{proof}
Suppose that there exists a flip $\phi^+:\,X^+\longrightarrow Y$ of $\phi$.
Since $\phi^+$ is a small contraction,  for any $l\geq 0$ we have $\phi^+_*(\mathcal{O}_{X^+}(lK_{X^+}))=\mathcal{O}_Y(lK_{Y})$. In particular, $\mathcal{A}=\oplus_{l\geq 0}\phi_*\mathcal{O}_{X^+}(lmK_{X^+})$. But $mK_{X^+}$ is $\phi^+$-ample (because for any contracted curve $C$ of $X^+$, $K_{X^+}\cdot C>0$), hence $\rm{Proj}(\mathcal{A})$ is a finitely generated sheaf of $\mathcal{O}_Y$-algebra and $X^+=\rm{Proj}(\mathcal{A})$.\\

Suppose now that $\mathcal{A}$ is finitely generated as sheaf of $\mathcal{O}_Y$-algebras, and define $X^+:=\rm{Proj}(\mathcal{A})$. Denote by $\phi^+$ the corresponding morphism $\rm{Proj}(\mathcal{A})\longrightarrow Y$. Then $X^+$ is clearly a normal variety.

For the rest of the proof, we choose $m$ sufficiently large such that $\mathcal{A}$ is generated by $\phi_*\mathcal{O}_X(mK_X)$, which equals $\mathcal{O}_Y(mK_Y))$; it does not change $\rm{Proj}(\mathcal{A})$. Then we denote by $\mathcal{O}_{X^+}(1)$ the $\phi^+$-very ample invertible sheaf on $X^+$ such that $\phi^+_*\mathcal{O}_{X^+}(1)=\phi_*\mathcal{O}_X(mK_X)$.

We know prove, by contradiction, that $\phi^+$ does not contract a divisor. Let $E$ be an irreducible exceptional divisor of $\phi^+$. We get the following exact sequence $$0\longrightarrow \mathcal{O}_{X^+}\longrightarrow\mathcal{O}_{X^+}(E)\longrightarrow\rm{Coker}\longrightarrow 0$$
where $\rm{Coker}$ cannot be zero. For a positive integer $l$, we apply the functor $\phi^+_*(-\otimes\mathcal{O}_{X^+}(l))$ to this sequence, to obtain $$0\longrightarrow \phi^+_*\mathcal{O}_{X^+}(l)\longrightarrow\phi^+_*(\mathcal{O}_{X^+}(E)\otimes\mathcal{O}_{X^+}(l))\longrightarrow\phi^+_*(\rm{Coker}\otimes\mathcal{O}_{X^+}(l))\longrightarrow R^1\phi^+_*\mathcal{O}_{X^+}(l)\longrightarrow\cdots$$

Choose $l$ sufficiently large such that $R^1\phi^+_*\mathcal{O}_{X^+}(l)=0$ and such that the map $(\phi^+)^*\phi^+_*(\rm{Coker}\otimes\mathcal{O}_{X^+}(l))\longrightarrow\rm{Coker}\otimes\mathcal{O}_{X^+}(l)$ is surjective (it is possible because $\mathcal{O}_{X^+}(1)$ is $\phi^+$-ample). Then $$0\longrightarrow \phi^+_*\mathcal{O}_{X^+}(l)\longrightarrow\phi^+_*(\mathcal{O}_{X^+}(E)\otimes\mathcal{O}_{X^+}(l))\longrightarrow\phi^+_*(\rm{Coker}\otimes\mathcal{O}_{X^+}(l))\longrightarrow 0.$$
We claim that the first map of the above sequence is surjective. Indeed, since $E$ is exceptional, for any open set $\mathcal{U}$ of $Y$, we have the following commutative diagram:

$$\xymatrix{
\mathcal{O}_X(lmK_Y)(\mathcal{U})\ar@{=}[r]\ar@{=}[d]  & \mathcal{O}_X(lmK_Y)(\mathcal{U}\backslash\phi^+(E)) \ar@{=}[d]\\
\phi^+_*\mathcal{O}_{X^+}(l)(\mathcal{U}) \ar@{^{(}->}[r] \ar@{^{(}->}[d] & \phi^+_*\mathcal{O}_{X^+}(l)(\mathcal{U}\backslash\phi^+(E)) \ar@{=}[d] \\
\phi^+_*(\mathcal{O}_{X^+}(E)\otimes\mathcal{O}_{X^+}(l))(\mathcal{U}) \ar@{^{(}->}[r] & \phi^+_*(\mathcal{O}_{X^+}(E)\otimes\mathcal{O}_{X^+}(l))(\mathcal{U}\backslash\phi^+(E)),
}$$
where all inclusions have to be equalities.

Hence, $\phi^+_*(\rm{Coker}\otimes\mathcal{O}_{X^+}(l))=0$. But  $\phi^{+*}\phi^+_*(\rm{Coker}\otimes\mathcal{O}_{X^+}(l))$ surjects to $\rm{Coker}\otimes\mathcal{O}_{X^+}(l)$, so $\rm{Coker}\otimes\mathcal{O}_{X^+}(l)=0$ and then $\rm{Coker}=0$. We get a contraction.

Now, since $\phi^+$ is a small contraction, we get that $\mathcal{O}(mK_{X^+})$ is isomorphic to  $\mathcal{O}_{X^+}(1)$ so that $X^+$ is clearly $\Qbb$-Gorenstein and $K_{X^+}$ is $\phi^+$-ample (ie for any contracted curve $C$ of $X^+$, $K_{X^+}\cdot C>0$).

Suppose now that $X$ has terminal singularities. We consider a common desingularization of $X$ and $X^+$:
$$\xymatrix{
 & V\ar[ld]_{\sigma}\ar[rd]^{\sigma^+}\\
 X\ar[rd]_{\phi} & &X^+\ar[ld]^{\phi^+}\\
 & Y &
}$$

Since $mK_{X^+}$ is $\phi^+$-ample, we have $$\begin{array}{rcl}
\sigma^{+*}\mathcal{O}_{X^+}(mK_{X^+}) & = & \sigma^{+*}(\Im(\phi^{+*}\phi^+_*\mathcal{O}_{X^+}(mK_{X^+})\longrightarrow\mathcal{O}_{X^+}(mK_{X^+})))\\
 & = & \Im(\sigma^{+*}\phi^{+*}\phi^+_*\mathcal{O}_{X^+}(mK_{X^+})\longrightarrow\sigma^{+*}\mathcal{O}_{X^+}(mK_{X^+}))).
\end{array}$$
But $\sigma^{+*}\phi^{+*}\phi^+_*\mathcal{O}_{X^+}(mK_{X^+})=(\phi^+\circ\sigma^+)^*\mathcal{O}_Y(mK_Y)=(\phi\circ\sigma)^*\phi_*\mathcal{O}_X(mK_X)=
(\phi\circ\sigma)^*(\phi\circ\sigma)_*\sigma^*\mathcal{O}_X(mK_X)$.
Thus, $\sigma^{+*}\mathcal{O}_{X^+}(mK_{X^+})$ is contained in $\sigma^*\mathcal{O}_X(mK_X)$.
In particular, by Lemma~\ref{lem:eff}, the divisor $\sigma^*K_X-\sigma^{+*}K_{X^+}$ is effective.

Hence, the divisor $K_V-\sigma^{+*}K_{X^+}=(K_V-\sigma^*K_X)+(\sigma^*K_X-\sigma^{+*}K_{X^+})$ is effective, and moreover, it has positive coefficient in the irreducible divisors of $V$ contracted by $\sigma$ (because $X$ has terminal singularities). It remains to prove that it has positive coefficient in the irreducible divisors of $V$ contracted by $\sigma^+$.

Let $E$ be an irreducible divisor of $V$ contracted by $\sigma^+$, but not contracted by $\sigma$ (if it exists). The irreducible divisor $\sigma(E)$ of $X$ is contracted by $\phi$. Then there exists a curve $C$  in $E$ that is contracted by $\sigma^+$ but not by $\sigma$. In particular, $\sigma(C)$ is a curve of $X$ contracted by $\phi$.
On the one hand $\sigma^{+*}K_{X^+}\cdot C=0$, and on the other hand $K_X\cdot \sigma(C)<0$ so that $\sigma^*K_X\cdot C<0$. Hence, $(\sigma^*K_X-\sigma^{+*}K_{X^+})\cdot C<0$. We can choose the curve $C$ such that it is contained in no other exceptional divisor of $\sigma$ and $\sigma^+$. Then we conclude as in the proof of Theorem~\ref{th:QGorDiv}, that the coefficient in $E$ of $\sigma^*K_X-\sigma^{+*}K_{X^+}$ is not zero.
\end{proof}

\begin{cor}[Corollary of the proof of Theorem~\ref{th:QGorFlip}]\label{cor:QGorFlip}
Let $\phi^+:\,X^+\longrightarrow Y$ be a flip of an extremal contraction  $\phi:\,X\longrightarrow Y$. Let $V$ be a commun desingularization of $X$ and $X^+$. Denote $\sigma:\,V\longrightarrow X$ and $\sigma^+:\,V\longrightarrow X^+$.
Then $K_V=\sigma^*K_X+\sum_{i\in I}a_iE_i=\sigma^{+*}K_{X^+}+\sum_{i\in I}a_i^+E_i$, such that, for any $i\in I$, $a_i^+\geq a_i$. Moreover, there exists $i_0\in I$ with $a_{i_0}^+>a_{i_0}$.
\end{cor}

Note that the set $\{E_i\,\mid\,i\in I\}$ is the union of the sets of irreducible exceptional divisors of $\sigma$ and $\sigma^+$, the hypothesis of Corollary~\ref{cor:QGorFlip} implies that for any $i\in I$, $a_i^+>0$, $a_i\geq 0$ and $a_i=0$ if and only if $E_i$ is not an exceptional divisor of $\sigma$.

\begin{proof}

It is enough to prove that, the effective divisor $\sigma^*K_X-\sigma^{+*}K_{X^+}$ is not zero.

Let $C$ be a curve of $V$ contracted by $\phi\circ\sigma=\phi^+\circ\sigma^+$ but not by $\sigma$. Then, by hypothesis on $\phi$, $\sigma^*K_X\cdot C<0$. If $C$ is contracted by $\phi^+$, we have $\sigma^{+*}K_{X^+}\cdot C=0$, and if not, we have $\sigma^{+*}K_{X^+}\cdot C>0$. In any cases,  $\sigma^{+*}K_{X^+}\cdot C\geq 0$ so that $(\sigma^*K_X-\sigma^{+*}K_{X^+})\cdot C>0$. In particular $\sigma^*K_X-\sigma^{+*}K_{X^+}$ is not zero.
\end{proof}

This corollary will be useful to prove the finiteness of sequences of flips in the family of $\Qbb$-Gorenstein spherical varieties.

\section{An example}\label{section4}

Here, we give an example of the $\Qbb$-Gorenstein MMP for a 3-dimensional toric variety. Se also \cite{Fujino3} for other toric examples.
For the basics of theory of toric varieties, the reader can see \cite{fulton} or \cite{oda}.

$\bullet$ In $\Zbb^3\subset\Qbb^3$, we consider the six following vectors:
$$\begin{array}{rclrclrcl}
e_1 &=& (-1,-1,1),\,&
e_2 &=& (1,-1,1),\,&
e_3 &=& (1,1,2),\,\\
e_4 &=& (-1,1,2),\,&
e_5 &=& (0,1,1),\,&
e_6 &=& (0,0,-1).
\end{array}$$

We denote by $\mathcal{C}(e_{i_1},\dots,e_{i_k})$ the cone in $\Qbb^3$ generated by $e_{i_1},\dots,e_{i_k}$. Then $$\mathbb{F}:=\{\mathcal{C}(e_1,e_2,e_3,e_4),\mathcal{C}(e_3,e_4,e_5),\mathcal{C}(e_1,e_2,e_6),\mathcal{C}(e_1,e_4,e_6),\mathcal{C}(e_2,e_3,e_6),\mathcal{C}(e_3,e_5,e_6),\mathcal{C}(e_4,e_5,e_6)\}$$ is a complete fan of $\Zbb^3$.
We denote by $X$ the toric variety of fan $\mathbb{F}$.

Note that since $\mathbb{F}$ contains a non-simplicial cone, the variety $X$ is not $\Qbb$-factorial.

Denote by $X_1$, $X_2$, $X_3$, $X_4$, $X_5$ and $X_6$ the $(\Cbb^*)^3$-stable irreducible divisors respectively associated to the rays of $\mathbb{F}$ generated by $e_1$, $e_2$, $e_3$, $e_4$, $e_5$ and $e_6$. We can compute that $$\rm{Pic}(X)_\Qbb:=\bigslant{\left\lbrace\sum_{i=1}^6a_iX_i\,\mid\,
\begin{array}{c} a_1-a_2+a_3-a_4=0\\
a_1,\,a_2,\,a_3,\,a_4,\,a_5,\,a_6\in\Qbb
\end{array}
\right\rbrace}{\left\langle \begin{array}{c}X_1-X_2-X_3+X_4,\\X_1+X_2-X_3-X_4-X_5,\\X_1+X_2+2X_3+2X_4+X_5-X_6\end{array}\right\rangle}.$$ In particular, $-K_X=\sum_{i=1}^6X_i$ is $\Qbb$-Cartier.

We know that $NE(X)$ is generated by the classes of $(\Cbb^*)^3$-stable (and rational) curves $C_{12}$, $C_{14}$, $C_{16}$,  $C_{23}$,  $C_{26}$, $C_{34}$ ,$C_{35}$, $C_{36}$,  $C_{45}$, $C_{46}$, $C_{56}$, where $C_{ij}$ denotes the $(\Cbb^*)^3$-stable curve associated to the 2-codimensional cone of $\mathbb{F}$ generated by $e_i$ and $e_j$.

We choose the basis $(X_1+X_2,X_1-X_3)$ of $\rm{Pic}(X)_\Qbb$ and we compute the classes of these curves in the corresponding dual basis:
$$\begin{array}{rclrclrclrcl}
[C_{12}] &=& (\frac{1}{3},0), &
[C_{14}] &=& (\frac{1}{6},0), &
[C_{16}] &=& (\frac{1}{2},0), &
[C_{23}] &=& (\frac{1}{6},0), \\
~[C_{26}] &=& (\frac{1}{2},0), &
[C_{34}] &=& (\frac{1}{3},\frac{1}{2}), &
[C_{35}] &=& (0,-\frac{1}{2}), &
[C_{36}] &=& (\frac{1}{2},\frac{1}{2}), \\
~[C_{45}] &=& (0,-\frac{1}{2}), &
[C_{46}] &=& (\frac{1}{2},\frac{1}{2}), &
[C_{56}] &=& (0,-1). &
\end{array}$$

In particular,  the cone $NE(X)$ is generated by $(2,3)$ and $(0,-1)$, ie by $[C_{34}]$ and $[C_{35}]$.

Note also that $-K_X$ is linearly equivalent to $5(X_1+X_2)-2(X_1-X_3)$ so that it is positive on all effective curves. Hence, in order to run the ($\Qbb$-Gorenstein) MMP, we need to choose one of the two extremal rays in $NE(X)_{K_X<0}$.\\

$\bullet$ First, consider the contraction $\phi:\,X\longrightarrow Y$ of the extremal ray generated by $[C_{35}]$. We remark that this ray contains the classes of $C_{35}$, $C_{45}$, and $C_{56}$. Then $Y$ is the 3-dimensional toric variety whose fan is $$\mathbb{F}_Y:=\{\mathcal{C}(e_1,e_2,e_3,e_4),\mathcal{C}(e_3,e_4,e_6),\mathcal{C}(e_1,e_2,e_6),\mathcal{C}(e_1,e_4,e_6),\mathcal{C}(e_2,e_3,e_6)\}.$$
We still denote by $X_1$, $X_2$, $X_3$, $X_4$ and $X_6$ the $(\Cbb^*)^3$-stable irreducible divisors of $Y$. And we compute that $$\rm{Pic}(Y)_\Qbb:=\bigslant{\left\lbrace\sum_{i=1}^4a_iX_i+a_6X_6\,\mid\,
\begin{array}{c} a_1-a_2+a_3-a_4=0\\
a_1,\,a_2,\,a_3,\,a_4,\,a_6\in\Qbb
\end{array}
\right\rbrace}{\left\langle \begin{array}{c}X_1-X_2-X_3+X_4,\\X_1+X_2-X_3-X_4,\\X_1+X_2+2X_3+2X_4-X_6\end{array}\right\rangle}.$$ In particular, $-K_Y=\sum_{i=1}^4X_i+X_6$ is $\Qbb$-Cartier (and generates $\rm{Pic}(Y)_\Qbb$). The contraction $\phi$ is a $\Qbb$-Gorenstein divisorial contraction. \\

$\bullet$ Now, consider the contraction $\phi:\,X\longrightarrow Y$ of the extremal ray generated by $[C_{34}]$. Then $Y$ is the 3-dimensional toric variety whose fan is $$\mathbb{F}_Y:=\{\mathcal{C}(e_1,e_2,e_3,e_4,e_5),\mathcal{C}(e_1,e_2,e_6),\mathcal{C}(e_1,e_4,e_6),\mathcal{C}(e_2,e_3,e_6),\mathcal{C}(e_3,e_5,e_6),\mathcal{C}(e_4,e_5,e_6)\}.$$
We still denote by $X_1$, $X_2$, $X_3$, $X_4$, $X_5$ and $X_6$ the $(\Cbb^*)^3$-stable irreducible divisors of $Y$.
And we compute that $$\rm{Pic}(Y)_\Qbb:=\bigslant{\left\lbrace\sum_{i=1}^6a_iX_i\,\mid\,
\begin{array}{c} a_1-a_2+a_3-a_4=0\\
a_1-3a_2+4a_3-6a_5=0\\
a_1,\,a_2,\,a_3,\,a_4,\,a_5,\,a_6\in\Qbb
\end{array}
\right\rbrace}{\left\langle \begin{array}{c}X_1-X_2-X_3+X_4,\\X_1+X_2-X_3-X_4-X_5,\\X_1+X_2+2X_3+2X_4+X_5-X_6\end{array}\right\rangle}.$$ In particular, $-K_Y=\sum_{i=1}^6X_i$ is not $\Qbb$-Cartier.

The flip of $\phi$ is given by the $\Qbb$-factorial 3-dimensional toric variety $X^+$ whose fan is \begin{multline*}\mathbb{F}_{X^+}:=\{\mathcal{C}(e_1,e_2,e_3), \mathcal{C}(e_1,e_3,e_5),\mathcal{C}(e_1,e_4,e_5), \\  \mathcal{C}(e_1,e_2,e_6),\mathcal{C}(e_1,e_4,e_6),\mathcal{C}(e_2,e_3,e_6),\mathcal{C}(e_3,e_5,e_6),\mathcal{C}(e_4,e_5,e_6)\},\end{multline*} and the $(\Cbb^*)^3$-equivariant map $\phi^+:\,X^+\longrightarrow Y$.
We still denote by $X_1$, $X_2$, $X_3$, $X_4$, $X_5$ and $X_6$ the $(\Cbb^*)^3$-stable irreducible divisors of $X^+$.
In the basis dual to $(X_1,X_2,X_3)$, the classes of the $(\Cbb^*)^3$-stable curves of $X^+$ are
$$\begin{array}{rclrclrclrcl}
[C_{12}] &=& (\frac{1}{3},0,\frac{1}{3}), &
[C_{13}] &=& (-\frac{1}{2},\frac{1}{2},-\frac{2}{3}), &
[C_{14}] &=& (\frac{1}{2},0,0), &
[C_{15}] &=& (-\frac{1}{3},0,\frac{1}{3}), \\
~[C_{16}] &=& (0,\frac{1}{2},0), &
[C_{23}] &=& (\frac{1}{6},0,\frac{1}{6}), &
[C_{26}] &=& (\frac{1}{2},0,\frac{1}{2}), &
[C_{35}] &=& (\frac{1}{3},0,\frac{1}{3}), \\
~[C_{36}] &=& (0,\frac{1}{2},-\frac{1}{2}), &
[C_{45}] &=& (1,0,0), &
[C_{46}] &=& (\frac{1}{2},0,0), &
[C_{56}] &=& (0,0,1).

\end{array}$$

 We deduce that the cone $NE(X^+)$ is generated by $(-3,3,-4)$, $(1,0,0)$ and $(-1,0,1)$. Moreover, since $-K_{X^+}$ is linearly equivalent to $3X_1+5X_2+2X_3$, there are two extremal rays in $NE(X^+)_{K_{X^+}>0}$, respectively generated by $[C_{13}]$ and $[C_{15}]$. The map $\phi^+$ is the contraction of the 2-dimensional face of $NE(X)$ generated by $[C_{13}]$ and $[C_{15}]$. \\

For more examples, we refer to \cite{MMPhoro} where a flip of a contraction that contracts a divisor is given.

\section{Log $\Qbb$-Gorenstein MMP}\label{section5}

As for $\Qbb$-factorial varieties, we can run a $\Qbb$-Gorenstein MMP for klt pairs $(X,D)$.

Let $X$ be a normal variety and let $D$ be an effective $\Qbb$-divisor such that $K_X+D$ is $\Qbb$-Cartier.

\begin{defi}
The pair $(X,D)$ is said to be klt (Kawamata log terminal) if $D$ there exists a desingularization  $\sigma:\,V\longrightarrow X$ of $X$ such that $K_V=\sigma^*(K_X+D)+\sum_{i\in I}a_iE_i$ where the $E_i$'s are irreducible divisors of $V$ and for any $i\in I$, $a_i>-1$.
\end{defi}

\begin{rem}
\begin{enumerate}
\item If a pair $(X,D)$ is klt, then the above property is true for every desingularization of $X$.
\item The condition "for any $i\in I$, $a_i>-1$" can be replaced by: $\lfloor D\rfloor=0$ and for any $i\in I$ such that $E_i$ is exceptional for $\sigma$, $a_i>-1$.
\end{enumerate}
\end{rem}

Suppose now that $(X,D)$ is klt.

By \cite{KMM}, the Contraction Theorem and the Cone Theorem are still valid. In particular, for any ray $R$ of $NE(X)_{K_X+D<0}$, there exists a unique contraction $\phi:\,X\longrightarrow Y$, such that, for any curve $C$ of $X$, $\phi(C)$ is a point if and only if the class of $C$ is in $R$. Moreover, we have equivalent results of Theorems~\ref{th:QGorDiv} and~\ref{th:QGorFlip}.

The following theorem was already given by O.~Fujino \cite{Fujino2} and generalized to log-canonical pairs in \cite{Fujino3}.

\begin{teo} We denote by $D_Y$ the $\Qbb$-divisor $\phi_*D$ of $Y$. And we fix $m\geq 1$ such that $m(K_X+D)$ is Cartier.
\begin{enumerate}
\item If $K_Y+D_Y$ is $\Qbb$-Cartier, then the pair $(Y,D_Y)$ is klt and $E:=K_X+D-\phi^*(K_Y+D_Y)$ is an exceptional and effective $\Qbb$-Cartier divisor such that $E.C<0$ for any curve $C$ of $X$ contracted by $\phi$.

\item The sheaf of $\mathcal{O}_Y$-algebras $\mathcal{A}:=\bigoplus_{l\geq 0}\phi_*\mathcal{O}_X(lm(K_X+D))$ equals $\bigoplus_{l\geq 0}\mathcal{O}_Y(lm(K_Y+D_Y))$.

\item If $K_Y+D_Y$ is not $\Qbb$-Cartier, $\mathcal{A}$ is finitely generated if and only if there exists a small contraction $\phi^+:\,X^+\longrightarrow Y$ such that the pair $(X^+,(\phi^+_*)^{-1}D_Y)$ is klt and for any curve $C^+$ of $X^+$ contracted by $\phi^+$, $(K_{X^+}+(\phi^+_*)^{-1}D_Y)\cdot C^+>0$. In that case, $X^+$ is $\rm{Proj}(\mathcal{A})$ over $Y$.
\end{enumerate}
\end{teo}

The proof is very similar to the proofs of Theorems~\ref{th:QGorDiv} and~\ref{th:QGorFlip}. The key of the proof of (1) is that, as in Corollary~\ref{cor:keyeff}, the divisor $K_X+D-\phi^*(K_Y+D_Y)$ is effective. Lemma~\ref{lem:key}, applied to $m(K_X+D)$ (and $(m(K_Y+D_Y)$) directly gives (2). And to prove (3), we do the same proof as in Theorem~\ref{th:QGorFlip}, by replacing $K_X$ by $K_X+D$, $K_Y$ by $K_Y+D_Y$ and $K_{X^+}$ by $K_{X^+}+(\phi^+_*)^{-1}D_Y$ (excepting the last paragraph, which is not necessary).

\section{Open questions}\label{section6}

The same questions as in $\Qbb$-factorial MMP can be done.\\

\noindent{\bf Question~1.} Do  flips always exist? Or equivalently, in the case where $Y$ is not $\Qbb$-Gorenstein, is $\mathcal{A}:=\bigoplus_{l\geq 0}\phi_*\mathcal{O}_X(lmK_X)=\bigoplus_{l\geq 0}\mathcal{O}_Y(lmK_Y)$ finitely generated as sheaf of $\mathcal{O}_Y$-algebras?\\

\noindent{\bf Question~2.} Are sequences of flips always finite?\\

We can answer positively these two questions in the case of spherical varieties.\\

Let $G$ be a connected reductive algebraic group (over $\Cbb$). A normal $G$-variety is spherical if there exists an open orbit in $X$ under the action of a Borel subgroup of $G$.

Let $H$ be a spherical subgroup of $G$ (ie such that there exists a Borel subgroup of $G$ satisfying that $BH$ is open in $G$). A $G/H$-embedding is a normal $G$-variety containing an open $G$-orbit isomorphic to $G/H$. (A $G/H$-embedding is a spherical $G$-variety, and inversely a spherical $G$-variety is a $G/H$-embedding for some spherical subgroup $H$ of $G$.)

\begin{prop} \label{prop:spher}
\begin{enumerate}
\item (\cite[Lemme 4.3]{brionmori}) Let $X$ and $Y$ be two spherical $G$-varieties and let $\phi$ be a proper $G$-equivariant morphism. Then for any Cartier divisor $D$ of $X$, the $\mathcal{O}_Y$-algebra $\bigoplus_{l\geq 0}\phi_*\mathcal{O}_X(lD)$ is finitely generated.
\item Let $X$ be a $G/H$-embeddings (ie a normal $G$-variety containing an open $G$-orbit isomorphic to $G/H$). There are only finitely many varieties $Z$ that can be obtained from $X$ by flips (and these varieties are still $G/H$-embeddings). Moreover, there exists a commun desingularization for all these varieties, ie there exist a smooth $G/H$-embedding $V$ and birational proper $G$-equivariant morphisms $\sigma_Z:\,V\longrightarrow Z$, for any $Z$.
\end{enumerate}
\end{prop}

The second part of the proposition is a consequence of the classification of $G/H$-embeddings in terms of colored fans and the fact that a flip adds no divisor.\\

Hence, in the family of spherical varieties, we immediately get the existence of flips, and the finiteness of sequences of flips is a consequence of {\it 2} of Proposition~\ref{prop:spher} and Corollary~\ref{cor:QGorFlip}.\\

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TITLE: Uniform convergence implies limit function is in $L^1.$
QUESTION [1 upvotes]: I have the following problem. Let $M>0$, and consider the sequence $(p_n)$ of real valued functions in $L^1([-M,M], \mathcal{B}[-M,M], \mu)$.  Suppose that $(p_n)$ converges uniformly to some $f: [-M,M] \rightarrow \mathbb{R}$. I need to show that $f$ is integrable. I thought of using the dominated convergence theorem, but I could not find a dominating $L^1$ function. Help is much appreciated :)

REPLY [1 votes]: Hint: Choose $n_0$ such that $|f-p_{n_0}|<1$ on $[-M,M].$ Use the triangle inequality to estimate $|f|$ in terms of $|p_{n_0}|.$


TITLE: How to calculate rental income of an appreciating property?
QUESTION [0 upvotes]: I'm trying to calculate the total rental income I might receive over 25 years. The rent is 5% of the property value, which is initially £300k. If the property itself increases in value by 3% per year, how much total rent will I have received after 25 years.
I'm trying to do this in 'Numbers' on a Mac, so if you can provide a function I can use for that, that'd be great. Thanks.

REPLY [0 votes]: You have a geometric series with starting value $5\% \text { of } 300=15$ and ratio of terms $1.03$, so the sum is $15\frac {1-1.03^{25}}{1-1.03}$


TITLE: Finding extremas of a function with only using derivative of the function.
QUESTION [0 upvotes]: $f'(x)=\dfrac{x^2(x-1)}{x+1}$ is given
Critical points are $0$ and $1$
Singular point is $-1$
$f''(x)=\dfrac{2x(x^2+x-1)}{(x+1)^2}$
So $-1$ is still not defined so it cannot be any extrema. Since function just diverges there.
For $x=1$ $f''(1)>0$ so it is local min.
About $x=-1$ is my assesment correct?

REPLY [1 votes]: No, it is not correct. Suppose, for instance, that $g(x)=\sqrt{\lvert x\rvert}$. Then it will be true that $g'$ and $g''$ are undefined at $0$. However, it is true that $g$ has a local minimum at $0$.
It is true, however, that your function $f$ does not have an extreme point at $-1$. This is so because, for some constant $K$,$$f(x)=2x-x^2+\frac13x^3-\log\left((x+1)^2\right)+K.$$


TITLE: Discrete Mathematics/SETS
QUESTION [1 upvotes]: Consider $G$ be an abelian group with $A$ and $B$ are subsets of $G$.
Consider $$A + B = \{a + b: a\in A, b \in B\},\quad A - B = \{a - b: a\in A, b\in B\}$$
If $A = B$, we have $A + A = \{a + a': a, a'\in A\}$ and $A - A = \{a - a': a, a'\in A\}$. For all $a, b$ distinct elements of $G$, we get $a - b$ is not equal to $a - b$ unless $a - b$ has order $2$.
Now the question is how to construct sets with more additions then subtractions (differences) for the finite set $A$ with $|A + A| > |A - A|$, where $|A|$ is a cardinality set of $A$.
Also, I want the proof of second part, For any set of integers $A$, for every integer $x$, how we define $x \cdot A = \{xa: a \in A\}$. If $A$ is set and can be expressible in more additions then subtractions for $x$ not equal to zero and for any integer $y$, the set becomes $x \cdot A + \{y\} = \{xa + y: a \in A\}$ is also an more additions than subtractions.
my final question is: How to prove the first explicit constructions of infinite families of (more additions than subtractions) sets of integers by using the probability method to prove the existence of such sets in certain finite abelian groups. in certain finite abelian groups?

REPLY [1 votes]: If you type "more sums than differences" into Google, you will be directed to several papers on this topic.


TITLE: Finding the area of a region?
QUESTION [3 upvotes]: I sketched out the graph to get this figure but I can't seem to find the area of the shaded region... would one Y = 4 and the other Y = 8?
I, in all honesty, am so flabbergasted with this question, any guidance will be much appreciated. I got 28/3 for the first integral (0 to 4) and 292/3 for the second integral (4 to 16)... I feel like I'm doing this wrong.
Okay, I'm going to be honest, can someone please solve this for me? I got 320/3 as my final answer and yet it's still wrong, I'm just about given up at this point, it's literally my last question of this semester.

REPLY [1 votes]: We want,
$$\left(\int_{0}^{4} \sqrt{x}\,\,dx-\int_{0}^{4}\dfrac12x\,\,dx\right)+\left(\int_{4}^{16}\dfrac12x\,\,dx-\int_{4}^{16}\sqrt{x}\,\,dx\right)$$
where the first term two terms give the area from $0$ to $4$ and the second term gives it from $4$ to $16$.
The expression becomes: $\left(\dfrac23x^{3/2}-\dfrac14x^2\right)\left.\right|_0^4+\left(\dfrac14x^2-\dfrac23x^{3/2}\right)\left.\right|_4^{16}$


TITLE: To show that a given graded function is the height function
QUESTION [1 upvotes]: My background in order theory is not very strong, and I'm confused about the following. The definition and theorems are from Lattices and ordered sets by Roman:
Let $(P,\le)$ be a partially ordered set, where $P$ is finite, i.e. $|P|=n$. Then we defined the distance $d(a,b)$ of two elements in $a,b\in P$ as the maximum length of all maximal chains from $a$ to $b$. Where a chain is defined to be a subset $S\subset P$ if $S$ is totally ordered by $\le$. In this finite case, we have just finite chains, which can be written in the form
$$a_1<a_2<\dots<a_n$$
where the length of such a chain is $n-1$. A chain from $a$ to $b$ is a chain in $P$, whose smallest element is $a$ and largest element is $b$. A maximal chain from $a$ to $b$ is a chain, which is not contained in a larger chain, in the sense of set inclusion, from $a$ to $b$. The distance $d(a,b)$ for elements $a,b\in P$ is defined as the maximum length among all maximal chains from $a$ to $b$. The height of a given element $a\in P$ is defined as $d(0,a)$ (we suppose $0\in P$).
I know that in a partially ordered set which contains $0$ it is equivalent to have the Jordan-Dedekind chain property and that $P$ is graded by its height function.
I have the following question: Suppose I know there is a function $g:P\to\mathbb{N}$ such that $a<b\Rightarrow g(a)<g(b)$. Moreover I know that $g(0)=0$. Now I was able to prove that for $a<b$ such that there is no $c$ with $a<c<b$ we have $g(b)=g(a)+1$. So I know that $P$ is graded by $g$. But why does it follow from $g(0)=0$ that $g$ is equal the height function, i.e. $g(a)=d(0,a)$?

REPLY [0 votes]: Let $C=(b_1=0,b_2,b_3, \ldots ,b_t=a)$ be a longest-length chain from $0$
to $a$. Then $d(0,a)=t-1$. Then there is no $c$ such that $b_i < c <b_{i+1}$
(otherwise $c$ could be inserted, producing a longer chain), so by the property
you have already shown, $g(b_{i+1})=g(b_i)+1$. It follows that $g(b_i)=i-1$ for
all $i$, hence $g(a)=t-1=d(0,a)$ and we are done.


TITLE: Is there a homotopy theory of unbased simply connected spaces?
QUESTION [10 upvotes]: If $\mathcal C$ and $\mathcal D$ are two categories each equipped with a class of morphisms called weak equivalences, then by an equivalence of homotopy theories I will mean functors $F:\mathcal C\to \mathcal D$ and $G:\mathcal D\to\mathcal C$ such that each of the compositions $G\circ F$ and $F\circ G$ is related to the relevant identity functor by a zigzag of natural weak equivalences. This is much stronger than just requiring $F$ and $G$ to induce inverse equivalences of homotopy categories: For an example, let $\mathcal C$ be the category of chain complexes over a field with quasi-isomorphisms as weak equivalences and $\mathcal D$ the subcategory of complexes with zero boundary map. ($F$ is the homology functor and $G$ is the inclusion.) You can see that this is not giving an equivalence of homotopy categories, because if it did then it would also give an equivalence of homotopy categories of $I$-diagrams for any $I$. Of course, if $\mathcal C$ and $\mathcal D$ are model categories then a Quillen equivalence gives an equivalence in the sense I mean (after restricting to categories of fibrant-cofibrant objects). But let's look at this simpler notion.
This is meant to be an improvement on my previous question When do basepoints matter in homotopy theory?. I was tempted to delete the latter, but decided to let it stand for its possible educational value and as a hairshirt for myself. I seriously suggest that nobody should upvote this question without downvoting that one.
Consider the category of simply-connected spaces (a.k.a. $1$-connected spaces, i.e those having exactly one path component and trivial fundamental group) and continuous maps. We can universally invert the weak homotopy equivalences and get what might be called the homotopy category of $1$-connected spaces. This can be described as a full subcategory of the homotopy category associated to the Quillen model structure on $Top$. Alternatively, it can be described as the category whose objects are $1$-connected $CW$ complexes and whose morphisms are homotopy classes of maps. Note that I am using unbased spaces and unbased maps here.
QUESTION 1: Is there an equivalence of homotopy theories in the sense defined above between $1$-connected spaces and (the fibrant-cofibrant objects of) some model category.
Another good example of a functor inducing an equivalence of homotopy categories but NOT an equivalence of homotopy theories is the forgetful functor from based $1$-connected spaces to unbased $1$-connected spaces. The positive statement just means that every unbased map between based $1$-connected spaces is homotopic to a based map, and that two such based maps are based homotopic if they are homotopic. The negative statement can be proved by using what I said above about categories of diagrams; for example, you can get a contradiction by picking an action of a group on a $1$-connected space such that the homotopy fixed point set is empty.
The category of $1$-connected based spaces is equivalent in this sense to a model category. You can colocalize the category of based spaces. The main idea is this: In the category of based spaces every object has a universal $1$-connected object over it.
As some kind of evidence for a "no" answer to the question above, note that the statement in the last sentence is false for unbased $1$-connected spaces. More evidence: a "yes" answer would seem to lead to a reasonable theory of homotopy limits and colimits within unbased $1$-connected spaces. How could that go? In the based setting, hocolim is usual hocolim and holim is universal covering space of basepoint component of usual holim.
All of the above statements, questions, and suggestions about $1$-connected spaces have analogues in which weak homotopy equivalence is replaced by rational equivalence. Sullivan's version
of rational homotopy theory shows that the theory of unbased $1$-connected spaces of finite type is equivalent to that of the opposite of $1$-connected commutative DGAs (differential graded algebras) of finite type. I presume that this can be used to see that the theory of (unbased) $1$-connected spaces is equivalent to that of $1$-connected commutative DGCs (differential graded coalgebras). Quillen's version gives an alternative route, but with basepoints: a chain of Quillen equivalences between based $1$-connected spaces and $1$-connected based (i.e. coaugmented) commutative DGCs (differential graded coalgebras).
QUESTION 2: Assuming the answer to Q1 is "no", do you have a good point of view to offer regarding this tradeoff between the artificiality of working with based objects and the disturbing lack of homotopy (co)limits in the unbased setting?

REPLY [9 votes]: It seems to me that the answer to (1) is "no", for essentially the reason you've given.
If $\mathcal{C}$ is a category with weak equivalences $W$, you can extract a simplicially
enriched category $\mathcal{C}[W^{-1}]$ using Dwyer-Kan localization, say. If this simplicially enriched category comes from a model category, then it has homotopy coproducts. That is, for every pair of objects $X$ and $Y$, there is another object $Z$ and a pair of maps
$X \rightarrow Z \leftarrow Y$ with the following property: for every object
$W$, the induced map Hom(Z,W) $\rightarrow$ Hom(X,W) $\times$ Hom(Y,W) is a weak homotopy equivalence of simplicial sets. The simplicially enriched category of simply connected CW complexes doesn't have these: for example, there is no homotopy coproduct of a point with itself.
As for $(2)$: you don't have arbitrary homotopy colimits, but you do have lots of them.
For example, you have all homotopy colimits indexed by diagrams with simply connected nerve.
This includes filtered colimits, geometric realizations of simplicial objects,
and the formation of pushouts. This is enough to allow for some useful techniques: for example, you can write every simply connected space as a geometric realization of a simplicial simply connected space, each term of which is a bouquet of $2$-spheres.


TITLE: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$
QUESTION [9 upvotes]: PROBLEM:  Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$
I know how to construct a diffeomorphism such that $\phi(p)=q$. One can see Guillemin and Pollack's book Differential Topology,page142, "Isotopy Lemma".
Alternatively, we can consider a compactly supported vector field $X$ with $\text{supp} X \subset U$ for some coordinate ball centered at $p$. WLOG, may assume $q$ lies in $U$ such that $q=(c,0,...,0)$ and $X= \partial_1$ on some $V \subset \subset U$. Now, since $X$ is compactly supported, the flow $\theta$ of $X$ is complete and we pick $\theta_c :M\to M$. Then,note that $\theta_c$ is a diffeomorphism and given by
$$(u^1,u^2,...,u^n) \to (u^1+c,u^2,...,u^n)$$ on some neighborhood of $p\in U$; hence, it is the desired diffeomorphism such that $\theta_c(p)=q$
However, with an additional requirement $d\phi(X_p)=Y_q$, I don't know how to continue. I try to adjust the above argument but fail to have any useful idea.

REPLY [2 votes]: (The idea behind this answer is essentially outlined in Neal's answer).
One, of course, assume that $X_p,Y_q\neq 0$, or there is nothing extra to prove.
First of all, let $T : M\to M$ be a diffeomorphism so that $T(p) = q$. Let $\psi: U \to \mathbb R^n$ be local coordinate at $p$. Then $T^{-1}\circ \psi : T(U)\to \mathbb R^n$ is a local coordinate of $q$, and $T =id : \psi(U)\to \psi(U)$ under these two coordinate. We assume that $T(U)$ is the unit ball $B_1(0)$ in $\mathbb R^n$ and $\psi(p) = 0$.
Under this identification, $X_p, Y_q$ are both vectors in $\mathbb R^n$ (rooted at $0 = \psi(p)$). Let $A : \mathbb R^n \to \mathbb R^n$ be an invertible linear transformation (with positive determinant) so that $A(X_p) = Y_q$. Let $r$ be small (depending on $A$), so that $A(B_r(0)) \subset B_1(0)$.
Let $B\in gl(n, \mathbb R)$ so that $e^B =  A$. Define a vector fields on $B_r(0)$ given by $Z(x) = Bx$. Now let $\rho : B_r(0)\to \mathbb R$ be a smooth cutoff function which is $1$ on $B_{r/2}(0)$. Then define a vector fields on $X$ given by
$$Q(x) = \rho(x) Z(x).$$
This vector fields can be treated as vector fields on $M$. Let $\alpha$ be the time one diffeomorphism corresponding to $Q$. As $Q(p) = 0$, we have $\alpha(p) = p$. Lastly, let $\phi = T\circ \alpha$. Then $\phi$ is a diffeomorphism and $\phi(p) = q$. To see $d\phi (X_p) =Y_q$, it suffices to check that in local coordinate $\psi: U \to \mathbb R^n$, $T^{-1}\circ \psi : T(U)\to \mathbb R^n$.
Under these coordinate, note that as $\rho$ is locally one, we have $\alpha_t (x) = e^{tB}x$ (locally on $B_{r/2}(0)$). In particular, in this coordinate,
$$\phi(x) = \alpha (x) = e^B x = Ax\Rightarrow d\phi_p = A\Rightarrow d\phi_p(X_p) = Y_q. $$


TITLE: Can a set have more than one ultrafilter?
QUESTION [0 upvotes]: https://proofwiki.org/wiki/Definition:Ultrafilter_on_Set
Given the definition that a filter $\mathcal F $ on a set $\mathcal S$ is  an ultrafilter on $\mathcal S$ iff there is no filter on  $\mathcal S$ which is strictly finer than $\mathcal F$, doesn't that just mean that the only ultrafilter on a set $\mathcal S$ is simply $\mathcal {P(S)} \setminus \{\emptyset\}$, the power set of $\mathcal S$ with the empty set removed?

REPLY [2 votes]: This is not a filter (it's not closed under intersection unless $S$ has one element).
Every set $S$ has at least $|S|$ many ultrafilters, namely, for any $s \in S$ the principal ultrafilter given by the subsets containing $s$. These are the only ultrafilters when $S$ is finite. When $S$ is infinite there are (assuming the axiom of choice) non-principal ultrafilters, which let you do lots of fun stuff.


TITLE: The limit $\lim\limits_{n\to\infty} (\sqrt{n^2-n}-n)$. Algebraic and intuitive thoughts.
QUESTION [3 upvotes]: I am working on the following problem.

Find the limit of $$\lim_{n \to \infty} (\sqrt{n^2-n}-n)$$

Intuitively, I want to say it's $0$ because as $n \to \infty$, $\sqrt{n^2-n}$ behaves like $n$ and subtracting $n$ makes it $0$.
However, algebraically, to my surprise
$$\begin{align}
\lim_{n \to \infty} (\sqrt{n^2-n}-n) & = \lim_{n \to \infty} \frac{n}{\sqrt{n^2-n}+n}\\
& = \lim_{n \to \infty} \frac{1}{\sqrt{1-\frac{1}{n}}+1}\\
& = \frac{1}{2} \\
\end{align}$$
Is there any intuitive way to explain why 0 was not the answer ?

REPLY [1 votes]: If you know that $\sqrt{1+x}\sim1+\frac x2$ for $x\sim0$, we get that
$$
\begin{align}
\lim_{n\to\infty}\left(\sqrt{n^2-n}-n\right)
&=\lim_{n\to\infty}n\left(\sqrt{1-\frac1n}-1\right)\\
&=\lim_{n\to\infty}n\left(\left(1-\frac1{2n}\right)-1\right)\\
&=\lim_{n\to\infty}n\left(-\frac1{2n}\right)\\
&=-\frac12
\end{align}
$$
or to complete your answer and correct the sign:
$$
\begin{align}
\lim_{n\to\infty}\left(\sqrt{n^2-n}-n\right)
&=\lim_{n\to\infty}\left(\sqrt{n^2-n}-n\right)\frac{\sqrt{n^2-n}+n}{\sqrt{n^2-n}+n}\\
&=\lim_{n\to\infty}\frac{(n^2-n)-n^2}{\sqrt{n^2-n}+n}\\
&=\lim_{n\to\infty}\frac{-n}{\sqrt{n^2-n}+n}\\
&=\lim_{n\to\infty}\frac{-1}{\sqrt{1-1/n}+1}\\
&=-\frac12
\end{align}
$$


TITLE: Different Sets of "K-up" Numbers
QUESTION [1 upvotes]: Recently I stumbled across a problem that is as follows:

Is it possible to find a number K such that there is a three-digit number that is K times the sum of its digits (a K-up number)?

I started by writing an equation:
$100a+10b+c=k(a+b+c)$
However, I couldn't find any relation between a, b, and c, or what will happen to them as k changes.
According to the given solution, there are 180 three-digit numbers that follow this rule. However, they are separated into different sets of numbers when K= a different number each time. For example, when
$K=37$:
$111=37(1+1+1)$
$222=37(2+2+2)$
$333=37(3+3+3)$
$...$
$999=37(9+9+9)$
There are 9 three-digit numbers that qualify when K=37, and $a=b=c$ (the three digits of the number)
Therefore, when K=37,
$a=\frac{3b}{7}+\frac{4c}{7}$
$b=\frac{7a}{3}+\frac{4b}{3}$
$c=\frac{7a}{4}+\frac{3c}{4}$
It may be time consuming to find one of these number by just randomly coming up with a number for K and another one for either a, b, or c.
My question is:

Is there a general rule for all values of K?
Is there a specific relationship between the digits a, b, and c as K changes?

REPLY [1 votes]: Not really an answer, but data.
Below is a complete list of three-digit "$k$-up" numbers (that is, numbers that when divided by the sum of their digits, yield $k$), sorted by $k$. If there's an overall pattern, I don't see it. (Note that the $37$-up numbers include more than the multiples of $111$.)
$$\begin{array}{c:l}
k \\
\hline
11 & 198 \\ \hline
12 & 108 \\ \hline
13 & 117, 156, 195 \\ \hline
14 & 126 \\ \hline
15 & 135 \\ \hline
16 & 144, 192, 288 \\ \hline
17 & 153 \\ \hline
18 & 162 \\ \hline
19 & 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 399 \\ \hline
20 & 180 \\ \hline
21 & 378 \\ \hline
22 & 132, 264, 396 \\  \hline
23 & 207 \\ \hline
24 & 216 \\ \hline
25 & 150, 225, 375 \\ \hline
26 & 234, 468 \\ \hline
27 & 243, 486 \\ \hline
28 & 112, 140, 224, 252, 280, 308, 336, 364, 392, 448, 476, 588 \\ \hline
29 & 261 \\ \hline
30 & 270 \\ \hline
31 & 372, 465, 558 \\ \hline
32 & 576 \\ \hline
33 & 594 \\ \hline
34 & 102, 204, 306, 408 \\ \hline
35 & 315 \\ \hline
36 & 324, 648 \\ \hline
37 & 111, 222, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 777, 888,
  999 \\ \hline
38 & 342, 684 \\ \hline
39 & 351 \\ \hline
40 & 120, 240, 360, 480 \\ \hline
41 & 738 \\ \hline
42 & 756 \\ \hline
43 & 516, 645, 774 \\ \hline
44 & 792 \\ \hline
45 & 405 \\ \hline
46 & 230, 322, 414, 460, 506, 552, 644, 690, 736, 782, 828, 874, 966 \\ \hline
47 & 423, 846 \\ \hline
48 & 432, 864 \\ \hline
49 & 441, 735, 882 \\ \hline
50 & 450 \\ \hline
51 & 918 \\ \hline
52 & 312, 624, 780, 936 \\ \hline
53 & 954 \\ \hline
54 & 972 \\ \hline
55 & 110, 220, 330, 440, 550, 605, 660, 715, 770, 825, 880, 935, 990 \\ \hline
56 & 504 \\ \hline
57 & 513 \\ \hline
58 & 522, 870 \\ \hline
59 & 531 \\ \hline
60 & 540 \\  \hline
61 & 732, 915 \\ \hline
64 & 320, 512, 640, 704, 832, 960 \\ \hline
67 & 201, 402, 603, 804 \\ \hline
68 & 612 \\ \hline
69 & 621 \\ \hline
70 & 210, 420, 630, 840 \\ \hline
73 & 511, 730, 803 \\ \hline
76 & 912 \\ \hline
78 & 702 \\ \hline
79 & 711 \\ \hline
80 & 720 \\ \hline
82 & 410, 820, 902 \\ \hline
85 & 510 \\ \hline
89 & 801 \\ \hline
90 & 810 \\ \hline
91 & 910 \\ \hline
100 & 100, 200, 300, 400, 500, 600, 700, 800, 900 \\ \hline
\end{array}$$