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

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2
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0answers
55 views

canonical bundle skew-symmetric determinantal varieties

By a skew-symmetric determinantal variety (over $\mathbb{C}$), I mean the projectivization of the set of skew-symmetric matrices of fixed size which rank is less than a given integer. More precisely, ...
1
vote
1answer
145 views

Zeroes of a complex polynomial on a sphere as a manifold

Let $ f \in \mathbb{C}[z_1, \ldots, z_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). ...
0
votes
1answer
74 views

Dominant map from hyperkahler manifolds to normal projective varieties with symplectic singularities

Let me recall some quick definitions. A projective hyperkahler manifold is a simply connected smooth projective variety $M$ such that $H^0(M,\Omega_M^2)=\mathbb C\sigma$, with $\sigma$ an everywhere ...
6
votes
0answers
110 views

Is there an non-finite etale map of varieties in char 0 with constant fiber size?

Let $k$ be a an algebraically closed field of char. $0$. Is there a morphism of varieties over $k$ which is: 1) etale 2) such that fibers at closed points all have the same size 3) yet not finite?
2
votes
0answers
124 views

Number of rational curves on varieties over finite fields

Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms over $k$. We define $$ \mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...
0
votes
0answers
82 views

Galois action on the components of a divisor

Let $F \hookrightarrow \mathbb{C}$ be a number field, say Galois. If $X$ is a smooth, quasi-projective variety over $F$, we can find by resolution of singularities a smooth, projective variety ...
4
votes
0answers
136 views

Dimension of a commuting nilpotent variety

Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
1
vote
2answers
131 views

a net of quadrics and the corresponding intersection

Let $Q_i(i=1,2,3)$ be quadric hypersurface in $\mathbb{P}^4$. Consider a net of quadrics $\Lambda=(Q_1,Q_2,Q_3)$. I can't understand some part of proof of Corollary 2.8(p.11) in Stability of genus 5 ...
1
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0answers
139 views

Are deformations of quotients of local rings embedded?

In Hartshorne's book "Deformation Theory" one can find a statement (inside the proof of Theorem 10.1) that every deformation $X' \to Spec(C)$ of an affine scheme $$X = ...
8
votes
0answers
200 views

degeneration of de Rham space “as” degeneration of crystalline site

Let $X$ be a smooth complex algebraic variety (for simplicity). Recall that the de Rham space (or stack) of $X$ is the quotient of $X$ by the groupoid that is the formal neighborhood of the ...
3
votes
0answers
174 views

Given a morphism of schemes, when does bijective + isomorphic tangent spaces = isomorphism?

Let $f: X \to Y$ be a morphism of schemes over a field $k$ such that $f$ induces (1) a bijection between their closed points, and (2) an isomorphism of their Zariski tangent spaces. Under these ...
7
votes
2answers
473 views

Hall-Littlewood functions and functions on the nilpotent cone

The following observation between the spaces of global sections of line bundles on the nilpotent cone and the Hall-Littlewood polynomials is made in a recent physics preprint 1403.0585. Is this a ...
1
vote
0answers
93 views

Submodul of finite ring extension

Let $R \hookrightarrow S$ be a finite extension of noetherian rings. Let $I \subseteq S$ be an $R$-submodule of $S$. Are there any sufficient criteria on $I$ such that it is in fact an ideal of $S$? ...
0
votes
0answers
67 views

moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc. Fix a non-negative integer $g$ and consider the space ...
5
votes
3answers
218 views

Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$: $$S_2\times Q\rightarrow Q,\; ...
12
votes
1answer
676 views

Why the name “variety” and the notation “V” for zeroes of polynomials?

The following questions came to my mind while preparing the notes for the first class of (my first) course on algebraic geometry. Question 1: Is there any motivation for choosing the term "variety" ...
10
votes
1answer
489 views

Several simple questions on the geometry of higher stacks

I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks. For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...
5
votes
0answers
47 views

Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
-1
votes
1answer
102 views

is the fixed point locus integral and reduced?

Let $X$ be a scheme over a field $k$ of characteristic zero and let $G$ be finite group acting on $X$. Then one can define the scheme $X^G$ of fixed points of $X$. It is a closed smooth subscheme of ...
1
vote
0answers
155 views

tensor of motives

Let $X$ and $Y$ be two smooth projective varieties over $k$. Consider the Chow motives $M(X\times\mathbb{P}^n)\simeq M(X)\otimes M(\mathbb{P}^n)$ and $M(Y\times\mathbb{P}^n)\simeq M(Y)\otimes ...
4
votes
1answer
174 views

Chow ring of two varieties

Suppose we are given two smooth projective varieties $X$ and $Y$. Maybe this is elementary but what is the Chow ring $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$?
1
vote
1answer
127 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
1
vote
1answer
92 views

endomorphisms of the Jacobian of a curve

Let $C$ be a smooth, projective curve over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. Precisely, ...
10
votes
2answers
394 views

Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
1
vote
1answer
152 views

blow up and derived category

Consider the blowup $X$ of $\mathbb{P}^2$ at a single point $p$. Then, Orlov showed that there is a semiorthogonal decomposition $D^b(X)=\langle e,O_X,O_X(1),O_X(2)\rangle$, where $O_X(i)$ is the ...
10
votes
0answers
140 views

real algebraic geometry software?

Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of ...
5
votes
1answer
178 views

Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post Normal form for a holomorphic Morse function Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...
1
vote
1answer
157 views

Is a semisimple conjugacy class closed?

Let $G$ be any algebraic subgroup of $\mathrm{GL}_n$ over an algebraically closed field of any characteristic. If $s$ is a semisimple element of $G$, can the $G$-conjugacy class of $s$ fail to be ...
7
votes
2answers
260 views

Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold

Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it being a member of a base-point-free linear system in a nef-Fano fourfold? What, in anything, is known ...
3
votes
0answers
126 views

Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?

Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...
5
votes
0answers
169 views

Kodaira vanishing for Du Bois singularities

Let $X$ be a projective variety with Du Bois singularities, which is additionally assumed to be Cohen-Macaulay. Then $H^i(X, \mathscr L^{-1}) = 0$ for any ample line bundle $\mathscr L$ and $i < ...
5
votes
1answer
487 views

Main conjecture for elliptic curves

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...
2
votes
1answer
148 views

Counting curves of degree 4 in $\mathbb{P}^{3}$

Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...
5
votes
2answers
278 views

Formal completion of the normal bundle

Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...
1
vote
1answer
126 views

Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
0
votes
0answers
67 views

cohomology of a space from a map to affine plane

Suppose $X$ is an affine variety,complete intersection in $\mathbb{C}^{2n}$, but with a high dimension of singularities. I also have a surjective finite algebraic map $f:X\rightarrow \mathbb{C}^{d}$. ...
5
votes
2answers
262 views

Making Hironaka's theorem explicit for hypersurfaces

Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, ...
0
votes
2answers
159 views

Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here: Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
2
votes
1answer
212 views

commutative diagram with Yoneda pairing, Weil pairing and edge morphism

Why does the following diagram commute?$\require{AMScd}$ \begin{CD} H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\ ...
0
votes
2answers
101 views

Poincaré bundle and Weil pairing for Abelian schemes

In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated. The same question for the Weil pairing ...
3
votes
1answer
135 views

$\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational

We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is ...
2
votes
1answer
104 views

Iterated extensions of quotients of vector bundles

Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties: Every locally free sheaf is in $\mathcal{S}$. If $A,B ...
3
votes
1answer
97 views

Normalization of a curve and push forward of vector bundles

Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. ...
1
vote
0answers
78 views

Positivstellensatz for non-polynomial term

Can we use Positivstellensatz (P-satz) below for a non-polynomial term? P-satz: Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...
2
votes
1answer
217 views

why do automorphisms preserve ample divisors?

Let $X \hookrightarrow \mathbb{P}$ be a smooth hypersurface inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$. ...
4
votes
2answers
364 views

Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has $$\otimes: D^{-}(X) \times D^{-}(X) \to ...
2
votes
1answer
169 views

geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

$R$ is a local Noetherian ring. What is geometric interpretation and of 1- Gorenstein rings 2- Complete intersections 3- Regular rings? how can I realize differences by geometric interpretation?
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vote
0answers
78 views

Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose $P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
3
votes
1answer
110 views

on a family of CM Hodge structures

I am having a look at the paper "Hodge structures of type $(n, 0, \ldots, 0, n)$" by Totaro. At the very beginning he says that a Hodge structure of type $(1, 0, 1)$ has always complex multiplication. ...
2
votes
2answers
172 views

Some questions on vanishing of Ext sheaves

Let $X$ be a complex manifold of dimension $3$ and $\mathcal{E}$ be a coherent sheaf such that $\dim supp(\mathcal{E})=1$. In this situation, I would like to know why we have the Ext sheaf ...