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

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1answer
73 views

Intersection product of pull back under projection

Let $X$ be a surface and $Y$ be a curve over $\mathbb{C}$. Let $L$ and $L'$ be ample line bundles on $X$ and $Y$ respectively. Consider the product $X\times Y$. Let $p$ and $q$ be the projection from ...
4
votes
0answers
235 views

Representable map of Deligne-Mumford stacks

Let $\mathscr{M}\to\mathscr{N}$ be a map of (Deligne-Mumford) stacks. Recall that it is said to be representable by affine schemes if for all affine maps $\operatorname{Spec}R\to \mathscr{N}$, the ...
1
vote
1answer
148 views

If the restriction of a vector bundle to a divisor is semi stable, then is the vector bundle itself semistable?

Let $X$ be a smooth projective variety of dimension $n$. Let $D$ be a smooth divisor of $X$. Let $i:D\hookrightarrow X$ be the inclusion. Let $H$ be an ample line bundle on $X$. Let $E$ be a vector ...
9
votes
2answers
458 views

Self-dual plane curves

Suppose that $C\subset \mathbb P^2$ is a plane projective curve (base field is $\mathbb C$) and $C^*\subset (\mathbb P^2)^*$ is its dual. What are the known examples in which $C$ is projectively ...
2
votes
1answer
191 views

Generalization of a theorem of Mahler to higher dimensions

A seminal theorem of Kurt Mahler, in his papers Zur Approximation algebraischer Zahlen. I-III., is the following: Let $F(x,y) \in \mathbb{Z}[x,y]$ be a binary form of degree $d \geq 3$ and non-zero ...
2
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1answer
127 views

necessary conditions for a quadric surface to be ruled (over a field of char 2)

Given a quadric surface $Q$ over a field $F$ of characteristic $2$, assume it is irreducible and reduced, we say it is ruled, if $Q$ is birational to $C \times \mathbb{P}^1$ for some $C$. A ...
6
votes
2answers
273 views

Global sections of coherent sheaves on determinantal hypersurfaces in $\mathbb{P}^n$

Let us consider the short exact sequence of coherent sheaves on $\mathbb{P}^n$ $$0 \to \mathcal{O}_{\mathbb P^n}(-1)^{r} \stackrel{N}{\longrightarrow} \mathcal{O}_{\mathbb P^n}^{r} \longrightarrow ...
3
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0answers
113 views

Equivariant Cohomology of flag varieties

Let $G$ be a simple simply connected algebraic group and $T$ be a maximal torus in $G$. Let $B$ be a Borel containing $T$ and $N(T)/T$ be the Weyl group. We have nice actions of $T$ and $W$ on the ...
3
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0answers
79 views

Is there a correspondence between counting curves in P^2 blown up at a point and counting curves in P^2?

Let $X$ be $\mathbb{P}^2$ blownup at one point and $\beta := d L -2E \in H_2(X, \mathbb{Z})$, where $L$ and $E$ denote the class of a line and the exceptional divisor respectively. Let ...
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0answers
24 views

Elimination theory for variables packaged in a matrix

I am wondering if the elimination theory in computational algebraic geometry can be more efficiently carried out if all variables lies within some given matrices. For instance, consider the following: ...
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109 views

Canonicity of Cech cohomology

For a topological space X, consider the Leray covering U_λ(i.e. ∩U_λ is sufficiently fine, e.g. affine for Zariski topology) of X. For a sheaf F on X, the cohomology H^i(X,F) is calculated via Cech ...
5
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142 views

Extension of sheaf of Azumaya algebras and derived equivalence

Suppose there is a smooth variety $X$ and a sheaf of algebra $\mathcal{B}$. Let $Z\subseteq X$ be a closed subvariety, whose codimension is large (say $\geq 2$). If the restriction of $\mathcal{B}$ to ...
5
votes
0answers
110 views

Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack

Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. ...
2
votes
2answers
355 views

Rational points on towers of curves

Let $\ldots \to X_n \to X_{n-1} \to \ldots \to X_0$ be etale maps between smooth projective curves of genera $g(X_n)>1$, all defined over a fixed number field $K$. By Faltings' Theorem, we know ...
15
votes
2answers
490 views

Grothendieck spectral sequence when one of the functors is contravariant

Let $f \colon X \rightarrow S$ be a morphism of schemes. I am interested in computing the cohomology groups of $$ \mathbf{R}\mathscr{H}om(\mathbf{R}f_* \mathcal{O}_X, \mathcal{O}_S) $$ in terms of ...
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votes
0answers
85 views

Sheaf of Sections of Cone

Fulton's intersection theory book at Chapter 4 makes the following claim: If $\mathcal{E}$ is a locally free sheaf (on $X$), and $E:=Spec(Sym(\mathcal{E}))$ a total space of some cone/bundle on $X$, ...
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0answers
108 views

Push-forwards of some codimension 2 classes from the universal curve to $\overline{\mathcal{M}}_{g,n}$

Let $\pi:\overline{\mathcal{M}}_{g,n+1}\to \overline{\mathcal{M}}_{g,n}$ be the map that forgets the last marked point and $\omega_\pi$ the relative cotangent line bundle (Here we are identifying ...
4
votes
1answer
134 views

Numerical invariants of symmetric products of curves

Let $n\ge 2$ be an integer and $C$ be a smooth projective curve of genus $g>n$. The $n$-the symmetric product $C(n)$ is a smooth variety of general type. If $n=2$ then $C(2)$ is a minimal surface ...
3
votes
0answers
174 views

Quartics containing twisted cubics

The set of quartic surfaces in $\mathbb{P}^3$ containing at least one twisted cubic forms the divisor on the space of all quartic spaces (parameterized by $\mathbb{P}^{34}$). I'am wondering how one ...
4
votes
2answers
251 views

Is the realtive dualizing sheaf Cohen-Macaulay?

Let $k$ be an algebraically closed field and let $X$ be a finite type $k$-scheme that is Cohen-Macaulay and equidimensional. Under these assumptions there is a relative dualizing sheaf $\omega_{X/k}$ ...
10
votes
3answers
386 views

Quotient of a smooth curve by a finite group and differentials

Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense ...
6
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0answers
108 views

Riemann-Roch for curves over Dedekind domains and base-change for modular forms

In p-adic properties of modular schemes and modular forms Katz formulates the following base change theorem as Theorem 1.7.1 Let $n\geq 3$ and $\overline{\mathcal{M}}_n$ be the compactified moduli ...
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0answers
88 views

When is the product of curves a complete intersection variety

Let $C$ be a smooth, projective curve over $\mathbb{C}$ of genus $g$. Let $L$ be a globally generated line bundle on $C$ and let $h^0(C,L)=r+1$. Let $\phi_L:C\rightarrow\mathbb{P}^r$ be the morphism ...
4
votes
1answer
161 views

Does derived equivalence of the fibres imply derived equivalence of the total spaces?

Let $f:X\to B$ and $g:Y\to B$ be smooth morphisms of complex projective varieties. Assume that for every closed point $b\in B$, the fibres $X_b=X\times \kappa(b)$ and $Y_b$ are derived equivalent. ...
2
votes
0answers
53 views

Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
2
votes
0answers
204 views

Why is the normalization of a general fiber the general fiber of the normalization?

Suppose $X \rightarrow Y$ is a map of reduced connected projective schemes of finite type over an algebraically closed field of characteristic 0, where $Y$ is a smooth connected curve. Let $Z ...
1
vote
1answer
103 views

Derived equivalence of families of dual abelian varieties

Let $B$ be a smooth projective complex variety and $\pi:X\to B$ a smooth projective map whose fibres $X_b$ are abelian varieties. Let $\psi:Y\to B$ be the naturally associated bundle such that the ...
4
votes
0answers
145 views

Does the cohomology comparison part of GAGA hold over the reals?

If $k=\mathbb{C}$, $X$ is a projective (or even proper) scheme over $k$ and $F$ a coherent sheaf on $X$, then there is an isomorphism $$ H^p(X,F)\rightarrow H^p(X^{an}, F^{an}) $$ (and there is also ...
0
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0answers
52 views

Non-local differentially smooth algebra

Let $A$ be a noetherian commutative algebra over a perfect field $k$. The algebra $A$ is said to be differentially smooth over $k$ if (1) $\Omega^1_{A/k}$ is a projective $A$-module, and (2) the ...
5
votes
2answers
200 views

Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...
3
votes
3answers
210 views

Is $Pic^0(X)$ of a curve of genus $\geq 1$ over a non-algebraically closed field still non-finitely generated?

Qing Liu's "Algebraic Geometry and Arithmetic Curves" page 299 COrollary 7.4.41 gives the following result. Let $X$ be a smooth, connected, projective curve over an algebraically closed field $k$, of ...
27
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0answers
659 views

Grothendieck's “List of classes of structures”

In Lawvere's article Comments on the Development of Topos Theory, the author writes: Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
3
votes
1answer
354 views

Confusion about a result on Shimura and Teichmüller curves

It is shown by M. Moeller (M. Moeller, Shimura- and Teichmüller curves) that there are only 2 Shimura and Teichmüller curves in the moduli space of curves $M_g$, namely, the ones given by ...
6
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0answers
137 views

Bogomolov-Beauville-Fujiki form, algebraically

Let $M$ be a compact hyperkahler manifold, i.e. a manifold with three complex structures $I,J,K$ defining an action of quaternions on the tangent bundle and a metric which is Kahler with respect to ...
3
votes
1answer
160 views

Maximal ideals of polynomial ring containing a fixed element

We know that for a field $k $ and $f\in k [x]$, the only maximal ideals of $k [x]$ containing $f $ are the ideals generated by prime factors of $f $. Now, I want to know that if $R $ is an arbitrary ...
3
votes
1answer
194 views

Existence of pencils on some special curves of genus 10

Everything over $\Bbb{C}$. Say we have a smooth curve $C$ of genus $10$ which is a double cover of a smooth plane cubic curve. Therefore $C$ admits a 1-dimensional family of pencils of degree 4 ...
14
votes
1answer
585 views

Is there a Serre intersection formula in analytic geometry?

There is the famous Serre intersection formula in algebraic geometry using the Tor functor (see for example here). I would like to know if there is such a formula in analytic (i.e. complex) geometry. ...
1
vote
1answer
91 views

Conormal bundle of the strict transform

I'm trying to understand the proof of Theorem II.1.7 in Kollar's Rational curves on algebraic varieties. In particular, there is a claim in there that I can't make sense of. The setting is the ...
6
votes
2answers
816 views

Does module Hom commute with tensor product in the second variable?

Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that $$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$ as $A$-modules? (Note that there is a ...
7
votes
2answers
205 views

The Picard number of the Kummer surface of an abelian surface

Let $A$ be an abelian surface and $\text{Km}(A)$ be the Kummer surface of $A$. If I remember correctly, the Picard number $\rho(\text{Km}(A))$ is equal to $16+\rho(A)$. Does anyone know any ...
4
votes
1answer
95 views

$Ex(f)$ has codimension at least 2

The following is a part of proof of lemma 6.2 in the book. $f:X \to Y$ a projective birational morphism of normal varieties $D$: Weil divisor on $Y$, $E$: exceptional divisor of $f$ ...
2
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0answers
96 views

Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$. Let $L\subset H$ be a ...
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0answers
87 views

Any natural examples of infinite dimensional Cohomological Field Theories?

Cohomological Field Theories as defined by Kontsevich and Manin are a class of linear maps from a vector space $V$ to the cohomology of the Deligne-Mumford moduli space of curves ...
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0answers
71 views

Irreducible real curves on ${\mathbb C}P^1$ invariant under the finite group action

Let $G$ be a finite subgroup of a Möbius group with a standart action on a real algebraic variety ${\mathbb C}{\mathbf P^1}.$ How one can describe $G$-invariant irreducible real algebraic curves? ...
2
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0answers
80 views

How the existence of holomorphic sections depends on the choice of complex structure

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...
3
votes
1answer
183 views

A Bertini-type result for hypersurfaces containing a subvariety

Let $P$ be a smooth projective variety of dimension $4$ and let $Z$ be an irreducible subvariety of dimension $2$ ($Z$ is not necessarily smooth, but you can assume it). Is there a smooth, ...
5
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0answers
105 views

Nonclassical polynomials, circles, and groups

Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm. A nonclassical polynomial of degree $d$ is a ...
3
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0answers
271 views

A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...
2
votes
0answers
101 views

Lifting group actions on a gerbe

Suppose $C$ is a smooth projective curve, over a number field. Let $G$ be its group of automorphisms. Suppose $\alpha$ is a class in $H^2(C,\mu_2)$ left fixed by $G$. Let $\mathcal{C}\rightarrow C$ be ...
6
votes
1answer
326 views

Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?