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

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138 views

Conditions for splitting of short exact sequence?

Are there conditions under which the short exact sequence $$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$ splits? I assume $K $ to be a number field and ...
14
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4answers
753 views

Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
1
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0answers
98 views

References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
0
votes
1answer
106 views

rationality of residues of differentials

Let $C$ be a smooth curve over a field $k$, $\overline{C}$ the smooth compactification and $S=\overline{C} \setminus C$. We think of $S$ as a reduced divisor defined over $k$. Take the sheaf of ...
2
votes
2answers
220 views

Theta characteristics of genus$\geq3$ curve

Let $C$ be a smooth curve of genus$\geq3$ over $\mathbb{C}$, so there are $2^{g-1}(2^g-1)$ odd theta characteristics and $2^{g-1}(2^g+1)$ even theta characteristics. Do we know how many of them has ...
4
votes
2answers
317 views

Specialisation of rigid varieties

Recall that a variety $X$ over a field $k$ is called rigid if $H^1(X, T_X) = 0$. I am interested in understanding this property under specialisation. Let $R$ be a discrete valuation ring and let ...
8
votes
1answer
389 views

Deformations of Ext rings

Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...
1
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1answer
137 views

Morphisms contracting a family of curves

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve. ...
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0answers
87 views

Looking for examples of holomorphic maps to $\mathbb{P}^1$ with certain property

I would like to know any example of nonconstant holomorphic map $f:X\to\mathbb{P}^1$ such that $K_X\cong f^*\mathcal{O}(2n)$ for some positive integer $n$, where $K_X$ is the canonical bundle of $X$. ...
3
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1answer
128 views

CM abelian varieties over the rationals

Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding ...
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41 views

Calculating the distinguished varieties of intersection product

In Fulton's Intersection theory Example 6.1.2,one considers two divisors on $\mathbf{P}^2$ given by $D_1=A+2B,D_2=2A+B$, where $A,B$ are lines meeting at a point. Let $X=D_1\times ...
0
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1answer
126 views

Sections of proper, flat morphism

Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is ...
1
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0answers
50 views

open set of hyperplanes not meeting a family of lines

Let \begin{align} \Omega=\begin{bmatrix} L_1 & \cdots & L_{n-1} \\ M_1 & \cdots & M_{n-1} \end{bmatrix}\end{align} be a matrix of linear forms on $\mathbb{P}^n$, i.e. homogeneous ...
3
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1answer
187 views

The stack of group algebraic spaces

The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see ...
2
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0answers
69 views

Surjectivity locus of a morphism of families of sheaves

Let $X$ and $T$ be schemes and assume we have two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X\times T$ which are flat over $T$, that is these are families of sheaves parametrized by $T$. ...
0
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0answers
63 views

a closure argument regarding certain linear functionals on polynomials

Let $S_d$ be the vector space of homogeneous polynomials of degree $d$ in two variables $x,y$ over an algebraically closed field $k$. Let $\phi \in S_d^*$ be a linear functional on $S_d$, such that ...
3
votes
1answer
170 views

Stacks with a small coarse moduli space

Let $k$ be a field of characteristic zero. Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$. Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$. ...
0
votes
1answer
144 views

Torsors and Central Extensions

In the setting of algebraic groups: I understand that a central extension of a group $G$ by an abelian group $A$ is a exact sequence of groups :$0\rightarrow A\rightarrow \tilde{G}\rightarrow ...
5
votes
1answer
250 views

Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless ...
2
votes
2answers
153 views

Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...
0
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1answer
234 views

Kummer theory and ramified covers

Let $X \to Y$ be a cyclic cover of algebraic varieties, with $Y$ smooth and $X$ normal, say over the complex numbers. Let $G$ denote the Galois group and let $\chi$ be a character. By Kummer theory ...
23
votes
2answers
649 views

How slowly can a power of an ideal grow?

For a polynomial ideal $I\subset \mathbb{C}[x_1,x_2]$, let $D(I)$ be the smallest degree of any polynomial in $I$. How slowly can $D(I^n)$ grow as a function of $n$? For example, if $D(I^n)\leq ...
1
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2answers
186 views

meaning of $k$-rational for closed subschemes

Let $X$ be a variety over a field $k$. I know the definition of $k$-rational point: a closed point $x$ is $k$-rational if its residue field $k(x)$ is equal to $k$ (in general, $k(x)$ is only a finite ...
2
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1answer
290 views

About normalization

Let $$f\colon X \to Y$$ be a morphism of affine normal algebraic varieties over $\mathbb{C}$. Assume that $f$ is birational and bijective on closed points. Does normality imply that $f$ is an ...
6
votes
1answer
312 views

Is “quotient” of projective variety projective?

Suppose $X$ is a projective variety, $f\colon X\to Y$ is a finite surjective morphism onto variety $Y$, must $Y$ be a projective variety?
10
votes
1answer
430 views

Which degree does a motivic Galois representation show up in?

Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...
4
votes
1answer
168 views

Does this extension of Hodge structures split over $\mathbb{Q}$?

Let $X$ be a smooth projective curve of genus $\geq 1$ over $\mathbb{C}$, $H^\cdot=H^\cdot(X)$, and $K$ be the kernel of cup product $\cup: H^1\otimes H^1\rightarrow H^2$. Consider the extension of ...
0
votes
0answers
105 views

Hodge numbers and weight filtration

Let $X$ be a complex smooth projective variety and $D$ a divisor on $X$ with normal crossings. As usual, denote by $D(m)$ the disjoint union of all possible intersections of $m$ irreducible components ...
0
votes
1answer
147 views

When is the normal neighbourhood of the boundary of the moduli space of cuvres parametrized by exactly one branch?

Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z}) $ a fixed homology class that is $\textit{decomposable}$. Let $$ \overline{\mathcal{M}}_{0,n}(X, \beta) $$ denote the stable ...
3
votes
1answer
236 views

Equivariant Derived Category

Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.) Suppose we have an algebraic group $G$ ...
2
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0answers
125 views

Projective bundle formula

Given a projective bundle $\mathbb{P}(E)\to X$, is there any reference for the projective bundle formula for Hodge cohomology ring $\oplus H^p(X,\Omega_{X/k}^q)$?
0
votes
0answers
116 views

Are fixed points of automorphisms rational?

Fix an integer $d$ and a number field $K$ containing the $d$-th roots of unity. Let $X$ be a variety over $K$ such that $Aut(X / K)$ has an element $\varphi$ of order $d$. Is it true that the fixed ...
0
votes
1answer
160 views

Grothendieck-Verdier duality for affine morphisms

Suppose $X,Y$ are varieties over $\mathbb{C}$, $Y$ is smooth and $X$ is Gorenstein ($X$ is not smooth in my case). Let $f: X \to Y$ be an affine morphism, and each fibre of $f$ has the same dimension ...
3
votes
1answer
191 views

Is the affine closure of a quasi-affine variety again a variety?

Question: Let $X$ be a quasi-affine scheme of finite type over a finite field $k$ and let $A:=\Gamma(X,\mathcal{O}_X)$. Under which conditions is it true that $A$ is a finitely generated $k$-algebra? ...
3
votes
1answer
213 views

Tate twist and comparison between Betti and de Rham cohomology

In Deligne's paper "Hodge cycles on abelian varieties" (see page 11 of http://jmilne.org/math/Documents/Deligne82.pdf) he says that the following diagram fails to commute by a factor of $(2 \pi i)^m$, ...
1
vote
1answer
126 views

Ext groups in the equivariant derived category

I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange. I am starting to learn about perverse sheaves, the ...
4
votes
0answers
160 views

Properties of schemes determined by field valued points [closed]

Are there any interesting cases (interesting here is interpreted rather loosely here) where you can show $X$ has property $P$ whenever all $X(K)$ have property $P$ where $K$ runs through all fields? ...
1
vote
0answers
144 views

Separability of a simple ring extension

Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
8
votes
1answer
329 views

Non-algebraic K3 surfaces in characteristic $p$

I have a very naive question. Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with ...
1
vote
1answer
157 views

Göttsche's formula for non-compact complex surfaces?

Is the Göttsche's formula (Eq (2.1) of this paper) expressing the Poincare polynomial (or the Euler char version) of the Hilbert scheme of points on a projective surface valid for non-compact complex ...
5
votes
0answers
151 views

Moduli of stable maps over the integers

Let $\overline{M}_{0,n}(X,\beta)$ be the moduli space of stable maps with target a smooth projective variety $X$. For which varieties $X$ and classes $\beta$ is $\overline{M}_{0,n}(X,\beta)$ well ...
1
vote
0answers
85 views

dimension of singular locus and complete intersection of a hypersurface

Let $X$ be a reduced projective hypersurface over a field $k$, which is defined by the homogeneous equation $f(T_0,\ldots,T_n)=0$. If the dimension of the singular locus of $X$ is $s$, $0\leq s\leq ...
4
votes
0answers
71 views

Effective Nullstellensatz and bounds on the nilpotency index of reduced ideal together with linear forms

Let $K$ be an algebraically closed field of characteristic $p>0$ and let $I\subset K[x_{1},\dots,x_{n}]$ be an ideal generated by (homogeneous) polynomials of degree $d$. Assume that $I$ is ...
4
votes
1answer
156 views

Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't ...
2
votes
1answer
91 views

sub-variety of (P^1)^4

Let $S$ be the sub-algebra generated by the set $\{x_1x_2y_3y_4, x_1x_3y_2y_4, x_1x_4y_2y_3, x_2x_3y_1y_4, x_2x_4y_1y_3, x_3x_4y_1y_2\}$ of homogeneous polynomials of degree $4$. I need to study the ...
0
votes
1answer
103 views

Formal Power series decomposition

Let $G$ be a linear algebraic group over $\mathbb C$ (say $SL_r$) consider a formal power series $$g(t)\in G(\mathbb C((t)))$$ My question is: Is it possible to decompose $g$ as $$g=ha$$ with $h\in ...
0
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0answers
129 views

How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?

This is a very basic question about the definition of Moduli space of maps. My reason for asking this question is because I haven't actually seen this definition explicitly given anywhere, and hence ...
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0answers
94 views

Moduli space of line-bundle holomorphic structures and sections over a Kahler-Hodge manifold

Let $(M,\omega)$ be a Kahler manifold with Kahler integral two-form $\omega$ and let $(L,h)$ be a rank-one complex vector bundle over $M$ equipped with a fixed hermitian metric $h$. I am interested in ...
2
votes
0answers
158 views

Unirationality and the Hasse principle

Is there an example of a quasiprojective variety $X$ defined over ℚ such that $X$ is unirational over all finite fields, and $X$ is unirational over $\mathbb{R}$, and $X$ is not unirational over ...
3
votes
1answer
199 views

Automorphisms of surfaces

Let $X$ be a projective surface with a morphism $f:X\rightarrow\mathbb{P}^1$. Assume that $f^{-1}(t)\cong\mathbb{P}^1$ for any $t\neq 0$ but $f^{-1}(0)$ is the union of two $\mathbb{P}^1$'s ...