Let $F \in \mathbb{Z}[x_0, \cdots, x_n]$ be a homogeneous polynomial. Let $V \subset \mathbb{P}^n(\mathbb{C})$ be a hypersurface (defined over $\mathbb{Q}$ say), given by a homogeneous polynomial $G(x_0, \cdots, x_n)$ say.

We say that $F$ *ramifies completely* on $V$ if there exists a positive integer $r > 1$ and polynomials $S,H$ such that $F(x_0, \cdots, x_n) = S^r + GH$. In other words, the image of $F$ with respect to the natural map $\mathbb{Z}[x_0, \cdots, x_n] \rightarrow \mathbb{Z}[x_0, \cdots, x_n)/I(V)$ is a perfect $r$-th power for some $r > 1$.

Unfortunately, assuming that $F$ is non-singular is not enough the exclude the possibility that $F$ ramifies on some hypersurface. Indeed, let $G(x_1, \cdots, x_n)$ be a non-singular polynomial of degree $d$ with respect to the variables $x_1, \cdots, x_n$, and put

$$\displaystyle F(x_0, \cdots, x_n) = x_0^d + G(x_1, \cdots, x_n).$$

It can easily be checked that $F$ is non-singular, and $F$ ramifies completely on the hypersurface given by $G = 0$.

Is there a way to classify those $F$ which ramifies completely on low degree hypersurfaces, namely hypersurfaces of degrees up to three?

Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the category of (oriented and abstract) simplicial complexes. An object in this category consists of a partially order set together with a collection of its subsets containing all singletons, close under taking subset, and such that each subset inherits a total order.

Is there an interesting combinatorially defined subcategory of cubical sets? Maybe one analogue to simplicial complexes?

It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.

This statement is usually called Poincaré duality.

One can also define $K$-theory with compact support (for sufficiently nice schemes $X$) by choosing a compactification $X \hookrightarrow \overline{X}$ and setting $K_c(X)$ as the homotopy kernel of $K (X) \rightarrow K (\overline{X}\setminus X)$. I have no idea on whether such $K_c$ and $K$ enjoy some kind of Poincaré duality.

When I hear something like Poincaré duality I expect some kind of cap product map with some fundamental virtual class $$H^{\bullet} \longrightarrow H_{d -\bullet}^{BM}$$ or, dually, $$H_c^{\bullet} \longrightarrow H_{d - \bullet}$$. Of course, there's a cap product $$K(X) \wedge G(X) \longrightarrow G(X)$$ induced by tensor product which when restricted to tensoring with $\mathscr{O}_X$ gives the Poincaré duality.

However, I'm not satisfied with such analogy. I, hence, ask the following.

1) Is there any sense in which $G$ is a $K$-theory with compact support? Or maybe it's even the opposite: $K$ is a $G$-theory with compact support?

2) If yes, is there any relation between $K_c (X)$ and $G(X)$?

3) If no, is there any kind of duality between $K (X)$ and $K_c (X)$?

4) If I'm actually sounding silly since in ordinary Poincaré duality both sides of the isomorphism are always simultaneously of the same kind (compact or not compact, for instance, $H_{\bullet}^{BM}$ is somehow non compact as $H^{\bullet}$), how can I see the duality as some isomorphism from a cohomology to a homology? In other words, why $K(X)$ should be a cohomology theory and $G(X)$ a homology theory?

5) If one uses some Atiyah-Hirzebruch spectral sequence for $G$-theory, would it be the case that the graded pieces of the $\gamma$ filtration define a motivic cohomology with compact support up to torsion?

6) What about 5 for $K_c$ instead of $G$? What about $G_c$?

7) After applying the Atiyah-Hirzebruch sequence to all the possibilities ($K$, $K_c$, $G$, $G_c$) what sort of Poincaré duality one acquires?

Thanks in advance.

**EDIT**

I've added new questions in order to correct my lack of attention to concordance of the "kind" (compact or noncompact) of the domain and codomain in the duality.

**EDIT2**

Given the comments below by Marc Hoyois and Gasterbiter, $K_c (X)$ should be defined as the homotopy colimit over $r$ of $K (\overline{X}, r (\overline{X}\setminus X))$, where the prefix $r$ denotes the infinitesimal thickening of order $r$ (following the notation of https://arxiv.org/abs/1211.1813).

Also, as noted below, $G$ should behave as a Borel-Moore homology. The analogy, therefore, is that

$$K(X) \longrightarrow G(X)$$ is the analogous of the first duality expressed above (cohomology-BM homology), whereas $$K_c(X) \longrightarrow G_c(X)$$ should correspond to the second duality (the compact version), where $G_c (X) := G (\overline{X}, \overline{X}\setminus X)$ (Btw, how do I state these dualities using the six functor formalism instead of using this "underline $c$"?).

Therefore, only the last questions remain. I will restate them here.

1) If one applies he Atiyah-Hirzebruch spectral sequence to $K$, $K_c$, $G$ and $G_c$, then what will be the graded pieces of the $\gamma$-filtration up to torsion (take $X$ as general as possible)? Or even better, in the level of spectra, what kind of decomposition one acquires?

For instance, in the case of smooth $X$, $K(X) \wedge \mathbb{S}_{\mathbb{Q}} \cong \bigvee_i H \mathbb{Q} \wedge (\mathbb{P}^1)^{\wedge i}$ (I have no idea what happens when $X$ is not smooth, though).

2) Does one recover some kind of Poincaré duality from the graded pieces mentioned in 1?

(This is a follow-up question to Positive real root separation)

Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not necessarily minimal).

**Additional assumption:** assume $\beta$ and $\gamma$ are the only Galois conjugates of modulus $>1$.

Numerically, there appears to be an absolute constant $C>0$ such that $|\gamma-\beta|\ge C$. Is this true/known? If it is, what is the best known value for $C$?

Furthermore, is it true that if the degree $d$ of $\beta$ is large, then $\min\{\gamma,\beta\}<1+\varepsilon$ and $\max\{\gamma,\beta\}>\frac{1+\sqrt5}2-\varepsilon$ with $\varepsilon\to0$ as $d\to\infty$?

This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the abstraction right.

A family of subsets of $\mathbb{N}$ is *centered* if every finite subfamily
has an infinite intersection. A *pseudointersection* of a family is an infinite set that is almost contained in every member of the family.

Let $\mathfrak{ridiculous}$ be the the minimal cardinality of a centered family of subsets of $\mathbb{N}$ with no 2 to 1 image that has a pseudointersection.

By *2 to 1 image* of a family $\mathcal{A}$
we mean the family $\{f[A] : A\in\mathcal{A}\}$,
for some 2 to 1 function $f\colon \mathbb{N}\to \mathbb{N}$.
($f[A]:=\{f(n):n\in A\}$).

We know that $\mathfrak{p}\le\mathfrak{ridiculous}\le \operatorname{add}(\mathcal{M})$. (We see this using selection principles; direct arguments of course must exist, too.)

**Question.** Is $\mathfrak{ridiculous}=\mathfrak{p}$?

A negative answer (i.e., consistently ``no'') would be ridiculous.

Recently, I am interested in the polynomial polynomial of the product of cycles. Let $G = (V , E)$ be an undirected multi-graph with vertex set $\{1,\cdots,n\}$. The graph polynomial of $G$ is defined by $$f_G(x_1,x_2,\cdots,x_n)=\prod_{1\leq i<j\leq n, (i,j)\in E}(x_i-x_j).$$

**Conjecture:** Let $G=C_{2n+1}\Box C_{2m}$, then the coefficient of $x_1^2x_2^2\cdots x_{(2n+1)(2m)}^2$ in the the graph polynomial $f_G(x_1,x_2,\cdots, x_{(2n+1)(2m)})$ is nonzero.

For $C_3\Box C_{2n}$, the conjecture is true. See the coefficient of a special term in the expansion of the graph polynomial

This conjecture generalized the result about $C_3\Box C_{2n}$. I think it may be true. But I have no idea about the proof on the general cases.

I hope someone could give suggestions about the conjecture. I will appreciate it even if given some special cases for the conjecture such as $n=2,3$,etc.

**EDIT: If the question is for SE level just delete from here as it is also posted there**. In fact I have seen some questions in SE regarding the coordinate rings of product of projective varieties but not the coordinate rings of intersection of hypersurfaces in products of projective varieties (maybe it is trivial, but I am not sure)

I am quite new in commutative algebra so I would be grateful if you could clarify to me this issue.

I would like to construct the coordinate ring, $A$, of a Calabi Yau threefold $X$ which is a complete intersection of two hypersurfaces $h_{1}$ and $h_{2}$ in the ambient space $\mathcal{A}=\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{2}$.

We can choose projective coordinates: $x=[x_{0}:x_{1}] $ for $\mathbb{P}^{1}$, $y=[y_{0}:y_{1}:y_{2}]$ for the first $\mathbb{P}^{2}$ and $z=[z_{0}:z_{1}:z_{2}]$ for the other $\mathbb{P}^{2}$. The configuration matrix is: (it is the Schoen manifold)

\begin{equation} X\left[\begin{array}{c||ccc} \mathbb{P}^{1}&1& 1\\ \mathbb{P}^{2}&0& 3\\ \mathbb{P}^{2}&3& 0\\ \end{array}\right]= \begin{cases} x_{0}f_{0}(y)-x_{1}f_{1}(y)=0& \\ x_{0}g_{0}(z)-x_{1}g_{1}(z)=0 & \end{cases} \end{equation} where $f_{0}, f_{1}, g_{0}, g_{1}$ are homogeoneous cubic polynomials.

I know that for a single projective factor, for example the quintic in $\mathbb{P}^{4}$ I can use $A=R_{\mathbb{P}^{4}}/I=\mathbb{C}[x_{0},x_{1},x_{2},x_{3},x_{4}]/\left ( \sum_{i=0}^{4} x_{i}^{5}+\psi x_{0}x_{1}x_{2}x_{3}x_{4} \right )$

**EDIT**
According to this the CICY case works in this way:
$\begin{gather}
\nonumber
A=\frac{R}{h_{1},h_{2}}
=(\mathbb{C}[x_{0},x_{1},y_{0},y_{1},y_{2},z_{0},z_{1},z_{2},]/ \left( x_{0}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{a} y_{0}y_{1}y_{2})-x_{1}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{b} y_{0}y_{1}y_{2}), \\ x_{0}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{c} z_{0}z_{1}z_{2})-x_{1}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{d} z_{0}z_{1}z_{2}) \right )
\end{gather}$

In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let $R = End(C)$, which is a maximal order in a quaternion algebra over $\Bbb Q$. Then he uses the fact that any projective module of rank $g \geq 2$ over $R$ is free by Eichler's theorem $[5]$.

The reference $[5]$ is an old paper written in German, so why is it free? More generally, what do we know about finitely generated projective modules (non only rank $1$ ones) over a maximal order inside some central simple algebras over a number field?

Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ be iid unit Exponential random variables, and let $(Y_k)_1^\infty$ be iid Exponentials with mean $v$ (i.e. $P(Y_1 \geq t) = e^{-t/v}$).

We are interested in if there is a closed form in terms of $p$ and $v$ for the probability $$P \left( \sum_1^\sigma X_k < \sum_1^{\sigma '} Y_k \right).$$

Conditioning on the value of either sum gives a messy expression that isn't obvious how to simplify.

A reformulation of the problem is to think of this as a race to reach 0 by two continuous time random walks with rates $1$ and $1/v$. Using the memoryless property, the probability the rate-1 walk advances at a jump time is $q=v/(1+v)$. Otherwise the rate-$1/v$ walk advances. Let $Z(r,q)$ be the number of successes before $r$ failures occur in iid trials with success probability $q$ (i.e. negative binomial). If we think of the rate-$1$ walk advancing as a success, we can rewrite the above probability as $$P( Z(\sigma,q) > \sigma').$$ Condition on the values of $\sigma$ and $\sigma'$ and use the distribution for a negative binomial to write this as $$\sum_{i,j \geq 0} C_i C_j p^2(p(1-p))^{i+j} \sum_{k\geq 2i+1} \binom{2j +k}{k} q^k (1-q)^{2j+1} .$$ Here $C_i$ is the $i$th Catalan number. It does not look easy to evaluate exactly. We are happier with this though because it is easier to numerically approximate (though we would prefer a closed form).

Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more general number fields? I was expecting to find it used in proofs of the functional equation for Dedekind L-functions attached to a number field, but I only found Poisson formulas for elements summed over an ideal rather than sums over ideals.

Is there anything of the form, for $K$ a number field, $E$ its ring of integer, and $f$ a suitable real valued function, $$\sum_{a \subset E} f(a) = C \sum_{a' \in E'} \hat{f}(a')$$

where the sum runs over integer ideals $a$ of $E$? What are the notions of dual and Fourier transforms here? Should we rather have something with norms of ideals instead of ideals? That is to say, is $N$ is the norm of the number field $K$,

$$\sum_{a \subset E} f(N(a)) = C \sum_{a' \in E'} \hat{f}(N(a'))$$

Thanks in advance, I do not find anything suitable in the litterature.

Let $\Delta$ be a simplicial complex on $n$ vertices, and $\phi$ a simplicial map that identifies two vertices $x$ and $y$ of $\Delta$. I want to show that the Betti numbers of $\phi(\Delta)$ cannot increase much from those of $\Delta$.

For instance, trivially $b_0(\phi(\Delta))\leq b_0(\Delta)$, and I could prove $b_1(\phi(\Delta))\leq b_1(\Delta)+1$. It is simply because $\mathrm{Ker}_{\phi(\Delta)}\partial_1\setminus\mathrm{Ker}_{\Delta}\partial_1=\phi\circ \partial_1^{-1}(\{c[x]-c[y]\mid c\in\mathbb{F}\})$. The same argument seems to only yield $b_k(\phi(\Delta))\leq b_k(\Delta)+O(n^k)$ for larger $k$.

So my questions are, are better results known for higher-order Betti numbers? Can we bound $b_2(\phi(\Delta))$ using both $b_2(\Delta)$ and $b_1(\Delta)$? Is it true that the sum of Betti numbers cannot increase much under identifying two vertices?

I've studied some fundamentals of algebraic geometry and number theory, and now I want to read papers which seem to be "main stream" of frontier research of arithmetic.

I've heard that Mazur's "Modular curves and the Eisenstein ideal" is one of such paper (and I've also heard that it is good for persons who have finished reading Hartshorne here), so I'm about to read it. But glancing through this, I feel that it goes far beyond Hartshorne, and that it needs the modular forms of moduli stack (I don't know what this is at all). And many papers of arithmetic seem to need this theory. However these papers refer to Katz's paper and Deligne, Rapoport's paper for the theory of modular forms, I feel these two papers are very difficult (and too long to start reading, not knowing that these are really good).

So my question is: please suggest me some references of the theory of modular forms (of moduli stack? Sorry, I know nothing, except that this modular forms are not one which I'm studying in Diamond, Shurman.). If Katz and Deligne, Rapoport are the best, or these are enough to read Mazur, I tried these.

Thank you very much!

Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row by the first $n^2$ elements of the Thue–Morse sequence with indexes from $0$ to $n^2-1$. Let $\mathcal D_n$ be the determinant of this matrix. For example, $$\mathcal D_7=\left| \begin{array}{ccccccc} 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ \end{array} \right|$$ For what $n$ the determinant $\mathcal D_n\ne0$?

I found $\mathcal D_2 = -1,\,\,\mathcal D_{11} = 9,\,\,\mathcal D_{13} = -9,\,\,\mathcal D_{19} = 270,\,\,\mathcal D_{23} = -900,$ and no other cases for $n<1000$. Are there any other cases?

Is it correct that

$\prod_{n : N}\prod_{m:N} n \leq m \rightarrow n < \text{succ}\ m$ ?

where the type $N$ of natural numbers has constructors

- $0:N$
- $\text{succ}:N\rightarrow N$.

(Hott Book: 5.1 Introduction to inductive types)

Say I have a square matrix $A$ of size $n\times n$. Then I tack on a column vector $\mathbf{u}$ of length $n$ on the right, and row vector $\mathbf{v}$ of length $n+1$ on the bottom. This creates a new square matrix $B$ of size $(n+1)\times(n+1)$. Is there a name for this type of transformation? How are the properties (e.g. determinant, inverse, rank, eigenvalues...) of $B$ related to the properties of $A$ and $\mathbf{u},\mathbf{v}$?

The fine Shafarevich-Tate (ST) group appears to not have been studied in much depth except for one paper by Wuthrich (2007). However, it only talks about fine ST group for elliptic curves, which makes me wonder if the results are in fact specific to elliptic curves or whether they can be generalised to Abelian varieties (specially something like Theorem 3.5 where it is proven that for elliptic curves over $\mathbb{Q}$ of rank 1, the $p$-primary part of the ST group and the fine ST group are equal.)

It is mentioned that it is believed for number fields in general, the fine ST group is expected to be small. But other than some examples, I did not see a moral reason to expect it.

Murty-Lim (2015) showed that the if one varies over $\mathbb{Z}/p$ extensions of number fields then the $p$-primary part of fine Selmer group of elliptic curves (their results are for Abelian variety) has unbounded growth. This combined with the results of Clark-Sharif from 2010, (where they proved that the $p$-torsion of the ST group of elliptic curves over number fields have unbounded growth in degree $p$ extensions) might suggest that $p$-primary part of the fine ST group can have unbounded growth.

Is there any recent work done on this?

Consider the mapping class group $MCG(\Sigma_2)$ of the closed genus 2 oriented surface $\Sigma_2$. The algebraic-duality theory of $MCG_2:=MCG(\Sigma_2)$ is explicitly described by Nathan Broaddus' very useful paper [https://arxiv.org/abs/0711.0011].

Broaddus constructs a beautiful homologically-nontrivial $2$-sphere $B$ in the curve complex of the genus two closed surface with marked curves α1, α2, α3, α4, α5, α6, β1, β2, β3 as labelled in the image below. ((we are indebted to N.Broaddus for his work and graphics.))

Question: I seek three distinct nontrivial elements $φ1,φ2,φ3$ of the mapping class group $MCG(\Sigma_2)$ with the property that the following chain sum vanishes over $\mathbb{Z}/2$-coefficients:

$α1 + α2 + α3 + α4 + α5 + α6 + β1 + β2 + β3$

$+ α1.φ1 + α2.φ1 + α3.φ1 + α4.φ1 + α5.φ1 + α6.φ1 + β1.φ1 + β2.φ1 + β3.φ1$

$+ α1.φ2 + α2.φ2 + α3.φ2 + α4.φ2 + α5.φ2 + α6.φ2 + β1.φ2 + β2.φ2 + β3.φ2$

$+α1.φ3 + α2.φ3 + α3.φ3 + α4.φ3 + α5.φ3 + α6.φ3 + β1.φ3 + β2.φ3 + β3.φ3$

By "vanishing over $\mathbb{Z}/2$-coefficients" we mean something utterly trivial like $\alpha+\beta + \alpha +\beta = 2 \alpha + 2\beta = 0$ (mod 2). Of course if we took $\phi_1 = \phi_2 =\phi_3=id$ or $\phi_1=\phi_2$, $\phi_3=id$ then the chain sum would vanish (mod 2).

Believe it or not, but finding such a triple of elements would yield a $MCG$-equivariant codimension-two spine $Z \hookrightarrow Teich(\Sigma_2)$ of the Teichmuller space.

The problem is related to stitching footballs from uniform hexagonal panels, or uniform pentagonal panels, or combinations of both. To stitch a football from panels $\{P_i| i\in I\}$ means finding a finite subset $I' \subset I$ for which the singular chain sum $\sum_{i\in I'} P_i$ has singular chain boundary which vanishes mod $2$, so $$\partial(\sum_{i\in I'} P_i)=\sum_{i\in I'} \partial P_i=0$$ over $\mathbb{Z}/2$-coefficients. When $P$ is two-dimensional hexagon or pentagon, the panels have singular boundary $$\partial P= \sum_{e\text{~edge~of~}P} e.$$ We denote the closed convex hull of the football $F:=conv\{P|\text{~panels}\}$. The panels then become closed subsets of the boundary $\partial F$. For instance since the 1960's, the standard football is stitched after Adidas' ``Telstar" design, having twenty white hexagon panels, and twelve black pentagon panels. But in our applications we assume the patches $\{P_i\}_I$ are pairwise isometric to some regular geodesically-flat polygon $P$.

Apologies in advance if this question is too elementary for MO. I asked it on the general math site earlier this week, but no one seemed to have an answer. I've been approximating answers via simulation, but I'd love a nice formula, and there must be one.

Suppose we have a biased coin that lands heads with probability *p* and tails with probability 1 - *p*. If the coin is flipped *x* times, what's the probability that there is some run of *n* consecutive flips in which the coin lands heads at least *m* times? (The *m* heads need not be consecutive, and of course *x* ≥ *n* ≥ *m*.)

By request, here's the original post. It isn't much different from this one, and got no responses.

I am reading a note at Page 63

ftp://ftp.math.ethz.ch/users/pink/FGS/CompleteNotes.pdf

It says whenever $G$ is finite and flat over $S$ the functor ${\rm Hom}(G,H)$ is representable. But it does not give the proof or any references.

Can anybody help me with this question?

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ and $u$ be a continuous convex function on the closure $\overline{\Omega}$. Given a boundary point $p\in\partial\Omega$, $u$ is said to have *infinite slop* at $p$ if $$\lim_{t\rightarrow0^+}\frac{u((1-t)p+tq)-u(p)}{t}=-\infty$$ for some $q\in\Omega$ (which implies the same limit for every $q\in\Omega$). Otherwise, $u$ is said to have *finite slop* at $p$.

On the other hand, the *Legrendre transform domain* of $u$ is the subset of the dual vector space $\mathbb{R}^{*n}$ where the Legendre transform of $u$ is defined (i.e. takes finite values). If $u$ is $C^1$ in $\Omega$, this domain is nothing but $\{Du(q)\mid q\in\Omega\}$.

The following question has an obvious positive answer when $n=1$, but I had some trouble trying to prove the "if" part in general:

**Question.** Is it true that for any convex function $u\in C^0(\overline{\Omega})$ with $u|_{\partial\Omega}=0$, the Legendre transform domain of $u$ is bounded if and only if $u$ has finite slope at every boundary point?

Is there a literature addressing this issue?