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

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Normal bundles of rational equivalent curves

Let $C_1, C_2$ be rationally equivalent curves in a smooth projective variety $P$. Let $$N_i: = \mathcal{H}om(I_{i}/I^2_{i}, \mathcal{O}_{C_i})$$ be the normal bundle of $C_i$, where $I_i$ is the ...
4
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1answer
153 views

liftings of principal bundles

I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise. Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for ...
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78 views

Deformation with fixed ramification

Suppose that $f : X \to Y$ is a finite, surjective morphism of normal varieties. I want to know about the space of first-order deformations of $X$ over $Y$ with fixed ramification, i.e. the ...
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1answer
81 views

Intersection multiplicty and global sections

Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between ...
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1answer
223 views

Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve”

I searched for it for a long time, but it seems that everybody is taking this for granted and does not bother to point out a proof. Would it be possible that someone points me to a proof or makes me ...
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1answer
368 views

Structure of the automorphism group of a Riemann surface

I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
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157 views

Adjunction map for projective surfaces

Before stating my question, let me recall (part of) the classical result on the adjunction map for complex projective surfaces, due in this modern form to Beltrametti and Sommese: Adjunction ...
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88 views

Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
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1answer
161 views

Hyperelliptic curve of genus 2 over R

I know that the points of an elliptic curve over $\mathbb{Q}$, $\mathbb{R}$ or other field $K$ form a group, particularly the most common example to explain the naive way is with this curve ...
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55 views

constant of functional equation of zeta function

Let $C$ be a smooth projective curve, of geometric genus $g$, over a finite field $\mathbb{F}_p$ and consider the zeta function $$ Z(C/\mathbb{F}_p, t)=\exp(\sum_{n=1}^{\infty} |C(\mathbb{F}_{q^n})| ...
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194 views

Fixed points of self maps

Given $m$ points on $S^n$, is there an explicit polynomial self $1-1$ map of minimum degree $f:S^n\rightarrow S^n$ that fixes only these $m$ points? Can we say something about symmetry group of $f$ if ...
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89 views

A birational compatification problem

Let $P_1$ be a projective variety over $\mathbb{C}$, and $Y_1 \subseteq P_1$ be a codimension $1$ (irreducible) subvariety. Suppose there is a blowdown morphism $Y_1 \to Y_2$. Then can I find a ...
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1answer
305 views

Exactness on rational points of algebraic groups

Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as ...
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2answers
292 views

Deformations of a blowup

Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let ...
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2answers
210 views

Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?

Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V ...
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1answer
332 views

Axioms for sheaf cohomology

Let $R$ be a commutative ring and $X$ a topological space. Define a sheafy cohomology theory (see here) to be a collection of functors $\mathrm{H}^q:\mathrm{Sh}(X;R\mathrm{Mod})\to R\mathrm{Mod}$ such ...
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1answer
218 views

Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...
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88 views

Real-rooted polynomials and higher rank matrices

For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + ...
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98 views

Pic^0 of the surface of bitangents of a quartic

Let $S$ be a generic quartic surface in $\mathbf{P}^3$. Let $T$ be the surface of the lines bitangent to $S$. What can we say about $Pic^0(T)$?
4
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1answer
188 views

Explicit bounds for transfer results in algebraic geometry

Assume you have an ideal $I\subseteq\mathbb{Z}[X_1,\ldots,X_n]$ of the polynomial ring in $n$ variables over the integers. For any field $\Bbbk$, I can consider the ideal ...
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1answer
185 views

Can Enriques Surfaces have non-trivial TWISTED Fourier-Mukai partners?

It is a well-known fact that for an Enriques surface $Y$, if $D^b(Y)\cong D^b(X)$ for some smooth projective variety $X$, then $X\cong Y$. In other words, Enriques surfaces have no non-trivial ...
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1answer
124 views

Hodge structure of relative cohomology groups

I need a hint or a good reference for definition of mixed Hodge structure on the relative cohomology groups ($\mathrm{H}^*(X,Y)$, $Y\subset X$ a closed subvariety of a comolex quasiprojective variety ...
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1answer
367 views

The difference between an étale finite group scheme and a finite group

I am trying to understand the statement that a Deligne-Mumford stack is locally a quotient $[U/G]$, where $G$ is a finite group. I don't understand why you can make $G$ a finite group, instead of a ...
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1answer
271 views

Non-proper intersection of surfaces

I'm interested in the first basic case of excess intersection in intersection theory: Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap ...
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1answer
113 views

How can one parametrize a real elliptic normal curve such that four points are coplanar iff their parameters sum to zero?

Let $E \subset \mathbb{P}^3_{\mathbb{R}}$ be a real elliptic normal curve with two non-null-homotopic connected components. Is there a parametrization $$ \chi: (\mathbb{R}/\mathbb{Z})\times ...
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136 views

Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?

Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...
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1answer
114 views

Extending holomorphic forms

Let $X$ be a normal variety over $\mathbb{C}$ and $\pi:\tilde{X}\rightarrow X$ a log resolution with (reduced) exceptional divisor $E$. Let $U$ be the smooth locus of $X$ and $\omega$ a holomorphic ...
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2answers
308 views

Line bundles over Kähler–Hodge manifolds

A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...
2
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1answer
139 views

Vanishing theorems involving symmetric powers of Kahler differential

Let $X$ be a smooth complex projective variety of dimension $d$ and $\Omega_X$ be its K$\ddot{\text{a}}$hler differential. Given an ample line bundle $L$ on $X$ and a positive number $k\ge 1$, denote ...
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309 views

$R^{\dim X-\dim Y}f_{\ast}\omega_X \simeq \omega_Y$ in positive characteristic?

In Proposition 7.6 of his paper "Higher Direct Images of Dualizing Sheaves", Kollár shows that if $X,Y$ are smooth complex projective varieties and $f:X\rightarrow Y$ is a proper surjective morphism ...
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98 views

Continuous isometries on Ricci flat compact manifolds

If I am not mistaken, a compact Ricci-flat manifold can have at most torus isometries. What is the name of the corresponding theorem or where can I find this result proven? It is known that ...
3
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2answers
492 views

orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$

Let $G\subseteq GL(n)$ be a linear algebraic group, and let $G({\Bbb Q}_p)\subseteq GL(V)$ act on a ${\Bbb Q}_p$-vector space V of finite dimension. Consider the action of $G$ on abelian subgroups ...
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66 views

Higher entry loci

Let $X\subset\mathbb{P}^n$ be a projective variety, and let $p\in\mathbb{P}^n$ be a general point. The secant cone $C_p(X)$ relative to $p$ is the union of all the secant line of $X$ through $p$. The ...
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223 views

Geometric vs combinatorial motives over Spec Z

Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
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1answer
203 views

An identity for Futaki-Donaldson invariant

Let $(X,L)$ be a polarized projective variety Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of : a scheme $\mathfrak X$ with a $\mathbb C^*$-action a ...
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100 views

Can we find a torus on $K^3$ surface

Suppose in $P^3$ we have $K3$ surface defined by $x^4+y^4+z^4+w^4=0$ can we find a complex subvariety that is a torus?
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174 views

Complex manifolds with trivial canonical bundle

It is known that a compact Calabi-Yau manifold can be defined as a compact Kahler manifold $M$ with trivial canonical bundle, or alternatively, a reduction of the structure group from $U(n)$ to ...
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102 views

Cohomology of pushforward under the double cover

Given a double cover $\pi: C \to \mathbb P^1$, where $C$ is a genus $g$ curve over algebraically closed field, I want to compute the group $\mathrm H^1(\mathbb P^1, \pi_*\mathbb G_m)$ in flat ...
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232 views

Cubic fourfold and K3 surface: geometric constructions of Hodge isometry

Hodge structure on K3 surface (the middle line of Hodge diamond is 1 20 1) is similar to the Hodge structure of cubic fourfold (the middle line of Hodge diamond of primitive cohomology is 0 1 20 1 0). ...
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191 views

The open problem of finding the explicit metric on a compact Calabi-Yau manifold

If I am not mistaken, no explicit metric on a compact Calabi-Yau manifold is known. I guess part of the difficulty is due to the fact that compact Calabi-Yau manifolds do not admit continuous ...
3
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1answer
355 views

Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? Open problem

I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer. Is the manifold $$M=\frac{E_{7(7)}}{SU(7)}\times ...
4
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0answers
134 views

Counting points on Hessenberg varieties over a finite field

Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...
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1answer
219 views

$\chi(\omega_X)>0$ implies that $X$ is of general type

If $X$ is a smooth (complex) projective variety of maximal Albanese dimension such that $\chi(\omega_X)>0$, how does one show that $X$ is of general type? I've seen this used but I can't find a ...
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1answer
170 views

An explicit formula for Weil pairing on a complex torus

I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers. Let ...
3
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2answers
310 views

What is $h^0(\mathcal O_F)$ where $F$ is a fiber of a normal surface over a smooth curve?

Lately I am studying the bend-and-break, and I follow the proof in the following note written by Olivier Debarre: http://www.math.ens.fr/~debarre/M2.pdf There is a detail that I just cannot go ...
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65 views

Linear independence of points under projection of Veronese re-embedding

Let $V$ be a complex vector space. Let $x_1,...,x_k\in PV$. Let $v_d: PV\rightarrow PS^dV$ be the Veronese. Then $v_d(x_1),...,v_d(x_k)$ are in general linear position as long as $k\leq d-1$. Now let ...
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3answers
204 views

Computing the coefficients of the polynomial $\dim H^0(X,L^k)$ in non-smooth case

Let $(X,L,\omega)$ be a projective variety with polarization $L$. then we can write $$\dim H^0(X,L^k)=a_0k^n+a_1k^{n-1}+...$$ If $X$ is smooth then $a_0=Vol(X)$ and we can compute $a_i$. If $X$ is ...
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80 views

How to construct a spherical bundle from a rigid curve on a threefold?

Let $X$ be a Calabi-Yau threefold. A vector bundle $E$ on $X$ is called spherical if $$ Ext^*(E,E)=H^*(S^3,\mathbb C). $$ Assume that a curve $C$ in $X$ is rigid and Brill-Noether general. ...
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1answer
109 views

Locally conformal Kahler manifolds with SU(4) structure

I would like to know if there exist eight-dimensional manifolds such that: It has SU(4)-structure. It is locally conformal Kahler. It is not a Calabi-Yau four-fold. A weaker question that also ...
2
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1answer
204 views

Self-intersection and generic point

The Wikipedia entry on intersection theory contains the following statement: [for C a curve, on a surface] "the self-intersection points of C is the generic point of C, taken with multiplicity C · ...