Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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-2
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55 views

toledo's lecture on cartwirght-steger surface

I am interested in Toledo's lecture given in IAS workshop. I want to find some related reference about his lecture. While actually i am not able to find much. Is someone also interested in this and ...
6
votes
0answers
99 views

scheme of commuting matrices

Let $k$ be any field. Let $r$ and $n$ be two positive integers. Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices ...
3
votes
2answers
131 views

Map between stacks and automorphism groups

I know that the Torelli morphism $t_g:\mathcal{M}_g\rightarrow \mathcal{A}_g$ between the stacks of smooth curves of genus $g$ and principally polarized abelian varieties of dimension $g$ is of order ...
2
votes
0answers
62 views

quotient a scheme by a stratified vector bundle

Let $k$ be a field. Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, ...
7
votes
2answers
228 views

Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
1
vote
0answers
76 views

Clarifications on twisted forms

Suppose $F = F(\bar{k})$ is a finite algebraic group over a number field $k$. The absolute Galois group $\Gamma_k$ of $k$ acts on $F$ by group automorphisms via a homomorphism $\rho: \Gamma_k \to ...
1
vote
0answers
105 views

Simply connected pencils

Is it true/false that for every complex projective variety $V$, there is a smooth simply-connected one parameter family $\pi:X\to \mathbb{P}^1$, one of whose smooth fibers is $V$? (i.e. we want ...
0
votes
0answers
65 views

About freeness of modules over the coordinate ring of an affine variety [migrated]

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...
0
votes
1answer
197 views

A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...
-2
votes
0answers
32 views

triangular inequality [on hold]

If $|a_n-L|\leq \epsilon$ and $|a_n-L'|\leq \epsilon$, then by the triangular inequality $|L-L'|\leq 2\epsilon$ I know that the triangular inequality says if $|a-b| \leq |a|+|b|$. However, I could ...
3
votes
1answer
148 views

Is $G \rightarrow G/P$ surjective on $K$-points over a local field?

Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?
1
vote
0answers
186 views

Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$. So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...
-2
votes
0answers
57 views

Stable curves and degenerations of smooth ones [on hold]

i'm approaching the study of Deligne-Mumford compactification for the moduli space of smooth curves of genus $g$. I'm trying to understand the geometrial meaning of stable curves: i know they have a ...
0
votes
2answers
148 views

Equidistribution of rational points on an algebraic variety

Suppose that we have a variety $X \subset \mathbb{P}^{n}$ defined over $\mathbb{Q}$. Suppose we have $S$ many rational points on $X$ inside the box defined by $|x_i| \leq B_i$ for $i = 0, \cdots, n$, ...
0
votes
1answer
125 views

Discriminant of a polynomial in two variables

I want to compute the discriminant of the following polynomial $$ F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j. $$ Here the discriminate means the equation $D(c_{i,j})$ in the variables ...
2
votes
0answers
128 views

Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf At page 10 he claims that an indirect ...
4
votes
1answer
88 views

Euler characteristic of open varieties as degree of Chern class of logarithmic differentials

Let $U$ be a smooth variety over a subfield $k$ of $\mathbb{C}$. Let $X$ be a smooth projective variety containing $U$ as the complement of a normal crossings divisor $D$. Denote by $\chi(U)$ the ...
0
votes
0answers
86 views

Can I find a resolution of singularities that is both smooth and projective? [on hold]

Let $X$ be a scheme of finite type over a field $k$. Can I find $X'$ that is both smooth and projective along with a birational, surjective, proper morphism $p:X'\longrightarrow X$? I have been ...
2
votes
1answer
53 views

Singularities induced by the toric ambient spaces

Let $\Delta \subset \mathbb{R}^4$ be a (reflexive) polytope and $X$ be the hypersurfacedefined by a generic section of the any-canonical bundle of the toric variety $\mathbb{P}_{\Delta}$. Are there ...
0
votes
0answers
103 views

Composition of multilinear forms agreeing on a subset of points

Let $n$ be a perfect square. Consider multiaffine polynomials $p(x_1,x_2,\dots,x_n),q(x_1,x_2,\dots,x_n),r(x_1,x_2,\dots,x_n),$$\{s_j(x_1,x_2,\dots,x_n)\}_{j=1}^{n}$$\in\mathbb R[x_1,x_2,\dots,x_n]$. ...
1
vote
1answer
135 views

Is the Cassels-Tate pairing defined for elliptic curves over function fields?

The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. ...
13
votes
1answer
246 views

Is the regularity of finitely generated rings decidable?

Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular? I mean by regular that the localization at every prime ideal is a regular local ...
1
vote
2answers
225 views

A question from the proof of affine algebraic group is a linear

In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", ...
0
votes
0answers
39 views

Semicubical parabola homeomorphic to C^2 [closed]

a semicubical parabola $L$ in $\mathbb C^2$ is given by $y^2=x^3$. I showed that a bijective function $f\colon\mathbb C \to L$ defined by $t \mapsto (t^2, t^3)$ becomes a homeomorphism regarding the ...
1
vote
0answers
114 views

Towards an enhanced version of homological mirror symmetry for affine varieties

Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories ...
2
votes
0answers
70 views

motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...
5
votes
0answers
90 views

Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind. Let $X$ be a smooth quasi-projective geometrically ...
0
votes
1answer
130 views

The linear projection of projective spaces

Let $\pi:X=\mathbb{P}_n\smallsetminus p\rightarrow \mathbb{P}_{n-1}$ be the linear projection. What is the cokernel of the morphism $$ 0\rightarrow \mathcal{O}_{\mathbb{P}_{n-1}}\rightarrow ...
2
votes
0answers
128 views

When is the proper transform of a subscheme equal to the total transform?

Say that $X \subset Y \subset \mathbb{P}^n$ are closed subschemes and let $\pi: \tilde{Y} \rightarrow Y$ be the blowup along $X$. If $H$ is a hyperplane section of $Y$, under what conditions (on ...
0
votes
0answers
17 views

Derivative of a projective transformation [migrated]

Assume $A$ is a matrix from $R^{n\times n}$, $A:R^n\rightarrow R^n$. Then $A$ induces a projective transformation $f:RP^{n-1}\rightarrow RP^{n-1}$. For example, $\\$ $$\begin{pmatrix} 4 & 0 ...
4
votes
0answers
118 views

Moduli interpretation of Hecke operators on Shimura curves

In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves. One can give definitions of the Hecke and Atkin-Lehner operators in terms of the ...
6
votes
1answer
115 views

For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS?

For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite ...
11
votes
2answers
490 views

The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get ...
2
votes
2answers
241 views

Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$

Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it is contained in a compact ...
0
votes
1answer
91 views

Is pushforward of an ample divisor under small birational map nef?

Let $X, Y$ be $\mathbb{Q}$-factorial, projective, normal varieties. Let $f: X --> Y$ be a small birational map. I have two related questions about pushforward of an ample divisor: (1) Let $H_X$ be ...
3
votes
2answers
503 views

Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold). Let $f_*\colon ...
0
votes
0answers
78 views

Books which defines higher differentials in algebraic curves context [migrated]

I'm reading an article which mentions a lot about higher differentials: I don't know what is $\Omega^n(F)$, my background is just Fulton's Algebraic curves book which defines just $\Omega(F)$. I ...
8
votes
1answer
405 views

Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?

Let $k$ be a field of characteristic $p > 0$ (algebraically closed, if you want; that doesn't make a difference). Consider a finite type $k$-group scheme $E$ that is a (central) extension of ...
0
votes
0answers
63 views

Looking for an example of linear algebraic groups with certain properties [closed]

I'm looking for an example of a linear algebraic group $G$ over a field $F$ of characteristic 0 such that $G$ is disconnected and not reductive (my defn of reductiveness doesn't require ...
0
votes
0answers
61 views

on lifting elements in a tangent space

Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$ Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and ...
0
votes
1answer
75 views

Vector bundle on ruled surface $X\times \mathbb{P}^{1}$

let $E$ be a vector bundle of rank $r$ on $X\times \mathbb{P}^{1}$ where $X$ is a smooth projective curve. Assume now that $E|_{F_{p}} \cong \mathcal{O}_{\mathbb{P}^{1}}^{r}$ for any p-fiber where $p: ...
1
vote
1answer
105 views

Some questions about ruled surfaces defined over $\overline{\mathbb Q}$

definitions: A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular ...
0
votes
0answers
73 views

Sufficient conditions to get complete intersection curves

Let $H_1,H_2\cdots,H_{d-1}$ be hypersurfaces in $\mathbb{P}^d$, if the intersection $B:=H_1\cap H_2\cap \cdots \cap H_{d-1}$ is $1$-dimensional then it is called a complete intersection curve. What ...
0
votes
1answer
192 views

Line on a hyper surface

Assume $X$ is a hyper surface in $\mathbb{P}^n$, can one always find a closed immersion $i:\mathbb{P}^1 \rightarrow X$?
2
votes
1answer
165 views

Number of minimal models of a surface

I would like to know if the following statement is true or false: Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism). ...
2
votes
1answer
225 views

When is the Hodge diamond concentrated in $H^{n,n}$'s?

Let $X$ be a smooth projective complex algebraic variety. The Hodge decomposition tells us that $H^n(X, \mathbf C) = \oplus H^{p,q}$. Here is my question: For what kind of $X$ is $H^{2n}(X) = ...
0
votes
2answers
119 views

Decompose a big divisor as nef big divisor and effective divisor

Let $W_n$ be a set of a log pair having the following property: For any $(X, D) \in W_n$ (1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a ...
1
vote
2answers
214 views

When the contraction is a morphism defined over $\overline{\mathbb Q}$

Suppose that $S$ is a complex projective surface defined over $\overline{\mathbb Q}$, namely there exists a surface $S_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that: ...
2
votes
1answer
150 views

About $\mathbb P^1_\mathbb C$ contained in a surface

Suppose that $X$ is a non-singular projective surface over $\mathbb {\overline Q}$ ( $X$ is a $\mathbb {\overline Q}$-scheme...) and suppose that there is an embedding: $$j:\mathbb P^1_{\mathbb ...
5
votes
0answers
140 views

Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...