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

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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 ...

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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 ...

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**1**answer

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 (...

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**1**answer

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 ...

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**1**answer

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
$$H^i(X,F(m))=0,\quad m>>0$$
...

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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 ...

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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 ...

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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 ...

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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 ...

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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^...

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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 $\...

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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 ...

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**1**answer

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 ...

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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 ...

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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 ...

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**1**answer

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 ...

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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?...

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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 ...

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**1**answer

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.. "...

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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 ...

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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 ...

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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}{\...

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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?

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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\...

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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 ...

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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 ...

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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 ...

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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?

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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 $\lambda$ ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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**1**answer

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)?

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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 ...

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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 ...

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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 ...

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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, ...

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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,*}(...

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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 ...

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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 $I_1$ ...

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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{...

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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.
...

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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 >...

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**1**answer

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 \...

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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 ...

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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 $$I_F=\left\{([l],[l'])\in F(X)\times F(X), l\cap l'\neq \emptyset\right\}$$ ...

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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 $\...

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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 ...