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

learn more… | top users | synonyms (1)

0
votes
0answers
23 views

Property of solid spanned by optimal functions

I can only solve this in a two-dimensional world. I want to characterize a solid that is spanned by cutoff functions $g$ that are optimal in the following sense. Let us suppose we have $n$ random ...
1
vote
0answers
216 views

On algebraic morphisms

Let given schemes $Y\subset X$ and $Z$, where $Y$ is closed subscheme of $X$. Assume that for some morphism $f:Y\to Z$, $Z = [f(Y)]$ (where [] means closure in $Z$). Is it true that there exists ...
10
votes
1answer
240 views

Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...
1
vote
1answer
145 views

Does a moving family of lines through a fixed point produce a singularity?

This is just a feeling that I had and I am curious if it is totally wrong or true to some extent. Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In ...
2
votes
1answer
133 views

Sequences of divisors satisfying Serre vanishing?

Serre's vanishing theorem (SV) states that, on a projective variety $X$ with a choice of ample line bundle $\mathcal{O}_X(1)$, for any coherent sheaf $F$, we have $$H^i(X,F(m))=0,\quad m>>0$$ ...
1
vote
0answers
87 views

Bounding the number of “generalized $\mathbb{F}_q$-rational points” of a variety in terms of dimension and degree

In what follows, for a prime power $q=p^m$, $\phi_q$ denotes the Frobenius endomorphism $x\mapsto x^q$ of a finite-dimensional affine space over the algebraic closure $\overline{\mathbb{F}_p}$ (the ...
4
votes
0answers
102 views

A sufficient condition for a morphism to be a closed immersion?

Let $R$ be an integral $\bar{\mathbb{F}}_p$-algebra of finite type, let $V$ be an $R$-algebra. Consider a morphism $f \colon \mathrm{Spec}(V) \rightarrow \mathbb{A}^n_R$ that has the following ...
3
votes
0answers
94 views

Is a Kummer surface unirational over a sufficiently large finite field of characteristic 2?

Let $A$ be a supersingular abelian surface over a sufficiently large finite field $\mathbb{F}_q$ of characteristic $2$ and let $K_A = A/(-1)$ be the Kummer surface. Shioda ("Kummer surfaces in ...
1
vote
0answers
117 views

Are two conic bundles birational, if their bases are birational via a map preserving the associated quaternion algebras?

Assume we have two standard conic bundles $\pi:C \rightarrow X$ and $\pi': C'\rightarrow X'$. That is $\pi$ and $\pi'$ are flat morphims of smooth varieties over $\mathbb{C}$ and both maps are ...
1
vote
0answers
96 views

GIT: For $x$ fixed, is $\{L:x \in X^s(L)\}$ open in $\text{Pic}^G(L)$?

Let $G$ be a complex reductive algebraic group acting on a complex variety $X$ (not necessarily projective) with $\text{Pic}^G(X)$ finite dimensional (for simplicity). For a fixed $x \in X$ define $$P^...
2
votes
0answers
69 views

Function field of a Drinfeld module, product formula, or Chow group

I am learning about Drinfeld modules, and I have a few questions. There is an analogue that Drinfeld modules are like elliptic curves, which are projective, or are compact Riemann surfaces over $\...
0
votes
0answers
87 views

is the “reduction” of an effective cartier divisor on a relative curve still a cartier divisor?

Let $C\rightarrow S$ be a smooth proper morphism of relative dimension 1, where $S$ is a Noetherian normal scheme. Let $D\hookrightarrow C$ be a relative effective Cartier divisor finite over $S$. Let ...
1
vote
1answer
133 views

Riemann-Roch for reducible surfaces

Let $C$ be a projective connected (reducible) curve over an algeraically closed field with nodes as singularities and $X=\mathbb P(\mathcal E)$ a projective bundle over $C$ (we know a ...
4
votes
1answer
383 views

Three dimensional representations of Alternating group

The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
9
votes
1answer
186 views

Geometric invariant theory and normalizers of stabilizers

For simplicity, work over an algebraically closed field of characteristic $0$. Let $$\begin{aligned} X &= \text{a smooth projective variety,} \\ G &= \text{a reductive group acting linearly on ...
6
votes
1answer
220 views

Generalised “scheme of etale connected components”

edit: Added smoothness condition. The following may be rather trivial, but I cannot seem to find a reference. If $X$ is a scheme write $et/X$ for the category of étale schemes over $X$ and $Sm/X$ for ...
1
vote
0answers
70 views

Constructive Resolution of Toric Singularities via Model Theory

Do there exists some language $\mathcal{L}$ of rational polyhedral cones in rational vector spaces and a theory $T$ over $\mathcal{L}$ whose models $\mathcal{M}$ are resolutions of toric singularities?...
2
votes
0answers
104 views

Image of a Weil divisor (height 1 prime) under normalization map

Let $R$ be a noetherian integral domain and let $R'$ be the normalization of $R$ in a finite field extension of the fraction field of $R$. Let $\varphi:Spec(R') \rightarrow Spec(R)$ be the ...
2
votes
1answer
174 views

Uniqueness of descent

Let $G$ be a reductive group, $X$ be a projective variety and $\mathcal L$ an ample $G$ equivariant line bundle on $X$. Then by a descent lemma of Kempf (see Narasimhan, M.S., and Drezet, J.-M.. "...
2
votes
0answers
211 views

Cohomology of intersection of projective hyperplanes

I will change my original question a bit for a bounty: Let $A$ be a reduced finitely generated $\bar{\mathbb{F}}_p$-algebra (integral, if you want). Let $X$ be a non-empty intersection of ...
1
vote
0answers
164 views

Is there a good notion of dual sheaf in a family?

Let $f:X\to S$ be a proper smooth morphism and let $E$ be a coherent sheaf on $X$ which is flat over $S$ and has codimension $c$. The scheme $X$ might not be smooth, as well as $S$ might not be ...
1
vote
0answers
136 views

Spivakovski-Popescu-Neron desingularisation

For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is, $A \cong \underset{\lambda \in \Lambda}{\...
1
vote
1answer
144 views

stable bundle on Calabi-Yau 3-fold

Let $E$ be a $G$-bundle over a compact Calabi-Yau 3-fold, $G$ is semi-simple,compact Lie group. If $E$ is a stable bundle, is the first Chern class of $E$ always zero?
5
votes
0answers
145 views

Fiber at infinity of an arithmetic surface $X$ as an element of $\widehat{\operatorname{Div}(X)}$

Introduction: Let $M$ be a Riemann surface, then a Green function on $M$ is an element $g\in C^\infty(V)$ where $V=M\setminus\{x_1,\ldots,x_r\}$ and around each point $p\in M$ we have: $$g=a\log\...
3
votes
1answer
205 views

Maximal Coset representative for the Weyl group of a Parabolic

Let $G=SL_n$ and let $P_i$ be a maximal parabolic corresponding to a simple root say $\alpha_i$. Let $W_{P_i}$ be the Weyl group of $P_i$. Is there an efficient way to compute the longest coset ...
8
votes
0answers
284 views

Semisimplicity of Frobenius on *integral* Tate module

Let $K$ be a number field and $A/K$ an Abelian variety; let $l$ be a (rational) prime. Do there exist infinitely many primes $\mathfrak{p}$ of $K$ such that the Frobenius at $\mathfrak{p}$ acts ...
12
votes
1answer
316 views

Rational curves on the Fermat quartic surface

Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...
1
vote
1answer
118 views

Divisor on variety determined by its restriction to curves

Is a (Cartier) divisor on a variety uniquely determined by its restriction to curves inside the variety? If so, how do we see this?
7
votes
1answer
229 views

Is a polarization on an abelian scheme an open condition?

Let $A/S$ be an abelian scheme such that the dual abelian scheme $A^{\vee}/S$ exists and let $\lambda : A \to A^{\vee}$ be a morphism of abelian schemes. Is the locus of points in $S$ where $\lambda$ ...
6
votes
1answer
294 views

Nonabelian $H^2$ and Galois descent

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety ...
0
votes
0answers
127 views

Localisation of the formal power series ring

Let $A \colon= K[[X_1,...,X_d]]$ be a formal power series ring of $d$-variables over a field $K$. Let ${\frak a}$ be a height $r$ prime of $A$ given by ${\frak a} \colon= (f_1,...,f_r)$, where $f_1 ...
2
votes
0answers
107 views

Double dual of ample sheaf

Let $X$ be a projective manifold. Then we can define ample sheaves on $X$, and many results of ample vector bundles still hold in this more general case (See K. Kubota, Ample sheaves). Now I was ...
4
votes
1answer
245 views

Elimination of noetherian hypothesis for abelian schemes

It is known that for every abelian scheme $A$ over a ring $R$, there exists a subring $R_0$ of $R$ that is of finite type over $\mathbb{Z}$ and an abelian scheme $A_0$ over $R_0$ such that $A$ is ...
3
votes
0answers
148 views

Relation between crystalline and perverse sheaves

Take $X$ to be a smooth complex projective algebraic variety. The Riemann-Hilbert correspondence gives an equivalence of categories between the category of perverse sheaves on $X$ and the category of ...
0
votes
1answer
229 views

Spectral sequences to compute Hom's in derived category

Does anybody have a good reference that lists spectral sequences that may be used to compute Hom sets in derived categories (of coherent sheaves, say)?
1
vote
0answers
70 views

Explicit construction of a bielliptic curve

Let $C$ be a (projective smooth complex) curve such that $K_C=2(D+p)$, with $D+p$ defining a $g_7^2$; $p$ is a base point and $D$ defines a 2-to-1 map $\varphi:C\rightarrow E\subset\mathbb{P}^2$ onto ...
2
votes
2answers
183 views

Reducibility of determinantal hypersurfaces

I have a determinantal hypersurface defined by $\det(A)=0$, with $a_{ij}$ homogeneous polynomial of fixed degree $d$ in $n$ variables. $A$ is not diagonal. How can I find out whether the hypersurface ...
1
vote
0answers
53 views

When the tensor product of motives that are not $0$-(homotopy)-connective can be $0$-connective?

For $t$ being the homotopy $t$-structure for Voevodsky effective motivic complexes when $M\otimes N\in DM_{eff}^{t\le -1}$ ensures that either $M$ or $N$ belongs to $DM_{eff}^{t\le -1}$ (so, we use ...
1
vote
1answer
76 views

Hyperplane generic to a given arrangement

At the moment, I am reading the paper "on the connectivity of the realization spaces of line arrangements" of Nazir and Yoshinaga. I would like to extend their Lemma 3.2 to higher dimension. However, ...
4
votes
1answer
190 views

Fourier Mukai transform for non-quasi coherent sheaves

Let $A$ be an abelian variety and $\hat A$ be the dual abelian variety. If $P$ is the (normalized) Poincare line bundle, then Mukai defines $R\hat S:D(A)\to D (\hat A)$ via $R\hat S(?)=Rp_{\hat A,*}(...
5
votes
1answer
278 views

Confusion regarding statement of mirror symmetry for elliptic curves

I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
-1
votes
0answers
124 views

An ideal and its annihilator

Let $I$ be an arbitrary ideal of commutative ring $R$ with identity. If $ann (I)=AB$, where $A$ and $B$ are comaximal, how can we fine two ideal $I_1$ and $I_2$ with $I=I_1+I_2$ such that $I_1$ ...
5
votes
0answers
119 views

Meaning of the support property in the definition of Bridgeland stability condition

Let $(Z,\mathcal{P})$ be a Bridgeland stability condition on a triangulated category $\mathcal{C}$. It is said to satisfy the support property if there exists a norm on $K(\mathcal{C})\otimes_\mathbb{...
2
votes
3answers
380 views

Krull-dimension of local domain

Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated. ...
2
votes
0answers
135 views

Inverse limit of Noetherian rings

Let $R_i$ be a noetherian regular local ring of Krull dimension being finite. Suppose we are given a surjective homomorphism $\phi_{i,j} \colon R_{j} \twoheadrightarrow R_i$ for each $i,j$ with $j >...
3
votes
1answer
177 views

Tensor and symmetric invariants of Symmetric group

For the action of $S_n$ on $\mathbb C^n$ the elementary symmetric polynomials generate the ring of polynomial invariants. What are the generators for the action of $S_n$ on $\mathbb C^n \otimes \...
4
votes
1answer
165 views

Resolution of the ideal of the Abel-Jacobi image of a curve?

Let $C$ be a complex curve of genus $g\ge 2$ and let $a\colon C\to J(C)$ be the Abel-Jacobi map. Is there a finite resolution of the ideal $\mathcal I_{a(C)}$ whose terms are sums line bundles of the ...
2
votes
0answers
94 views

Reference quest: variety of lines and variety of planes

Let $X\subset \mathbb P_{\mathbb C}^n$ be a smooth projective variety, $F(X)\subset G(2,n+1)$ its Fano variety of lines and $$I_F=\left\{([l],[l'])\in F(X)\times F(X), l\cap l'\neq \emptyset\right\}$$ ...
6
votes
0answers
269 views

Description of connecting maps of Derived functors

Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...
11
votes
0answers
452 views

Why should Algebraic Geometers and Representation Theorists care about Geometric Complexity Theory?

Geometric Complexity Theory has demonstrated that Complexity Theorists should care about Algebraic Geometry and Representation Theory, but, why should Algebraic Geometers and Representation Theorists ...