# Questions tagged [ag.algebraic-geometry]

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

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### Rational points on quartic surfaces

Let $S\subset\mathbb{P}^3_{K}$ be a surface of degree $4$ over a field $K$. Assume that $S$ has a double line also defined over $K$ say $L = \{x = y = 0\}$, where $x,y,z,w$ are the homogeneous ...
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### How do I find the center of mass of a region given by a function using integration [closed]

So I'm currently working on a project where i want to find the center of mass of a given region. To be exact, i want to find the center of mass (using moments and all of that) of this region the plane ...
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### A question on Springer's theorem

Springer's theorem in T. A. SPRINGER, Sur les formes quadratiques d’indice zéro, C. R. Math. Acad. Sci. Paris 234 (1952), 1517–1519. asserts that if a quadric fibration $\pi:X\rightarrow Y$ over a ...
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### Blowups of log del Pezzo surfaces at smooth points

It follows from a result of Küchle that the blowup of a smooth del Pezzo surface will again be del Pezzo, provided that the inequality $-K^2>0$ remains true after blowing-up. Let's say a surface is ...
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### Künneth theorem in étale cohomology

I am searching for an account of the Künneth theorem in étale cohomogy. Does the Künneth theorem in étale cohomology also follow from the 6-functor formalism or some other formalism? It would be nice ...
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### Can this embedding to double dual EPW sextic happen?

Let $\widetilde{Y}_{A^{\perp}}$ denote the double dual EPW sextic defined by the Lagrange subspace $A\subset \bigwedge^3V_6$. If $A$ is very general, then $\widetilde{Y}_{A^{\perp}}$ is a smooth ...
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### Algebraic correspondence as morphisms in Betti cohomology

$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
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### Hartshorne's proof of Halphen's theorem

Apologies if this is not quite at the level of MathOverflow, but it has already been asked at MSE and gone unresolved for several years despite a bounty. Hartshorne states the theorem as follows: ...
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### Hodge's conjecture as a quasi-isomorphism between two complexes of sheaves

A version of Hodge's conjecture due to Beilinson, expects that the Betti cycles class map $H_{\mathcal{M}}^i(X,\mathbb{Q}(j))\rightarrow hom_{MHS}(\mathbb{Q}(0),H^{i}(X,\mathbb{Q}(j) ))$ is surjective ...
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### Nearby cycle is tamely ramified?

Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
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### Second Chern class of a smooth projective variety

Suppose $X$ smooth projective variety of dimension $n$ such that $-K_X$ is ample. If $h^0(-K_X) >0$, then the first Chern class of $X$ can be seen as a cycle of co-dimension $1$ associated to a ...
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### The upper bounds on rank $2$ real matrices

Let $A_{n}(F)$ be the collection of all skew-symmetric matrices over the field $F$ ($\operatorname{char} F \neq 2$). Let M be a subspace of $A_{n}(F)$ such that all non zero elements have rank ...
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### Local description of the universal family $\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}$

I would like to get an understanding of the notion of geometric fibers of the universal family: $$\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}.$$ In fact Knudsen show ...
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### The Hodge number $h^{2,0}$ of (finite) quotient variety of a K3 surface

Let $X$ be an (algebraic) K3 surface, then we have $H^{2,0}(X)=\langle \omega_X\rangle$, where $\omega_X$ is the period. Suppose $G=\langle g\rangle$ is a finite group acting on $X$ and $g$ as an ...
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### How does intersection form on vanishing cohomology determine hodge type?

In the paper "Complete intersections with middle picard number 1 defined over Q" by Tomohide Terasoma (1985), page 295, line 7 from the bottom, we are in the following situation: We have a ...
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### Normal bundle of a Fano threefold as Brill-Noether loci

Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient ...
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### Does the first cohomology of the Hodge bundle over the moduli space of curves $M_{g,n}$ vanish?

I know the Leray spectral sequence relates this to the cohomology of the relative dualizing sheaf, but I don't know anything about the cohomology of either of these sheaves.
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### Does Lefschetz pencil always exist in char $p$?

Let $X\subset \mathbb{P}^n_k$ be a smooth projective variety, a point $p\in \mathbb{P}^{n,\vee}_k$ gives rise to a hyperplane $H_p\subset \mathbb{P}^n$, hence an intersection $X_p:=H_p\cap X$. We say ...
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### Is every Zariski closed subgroup a stabilizer?

Let $G$ be a linear algebraic group. Is it true that a subgroup $H$ of $G$ is Zariski closed if and only if there exists a representation $\pi: G \to \mathrm{GL}(V)$ and a vector $v \in V$ ...
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### Support of torsion in the Borel–Moore homology

Given a complex quasi-projective variety $X$, let $\alpha$ be an element of the Borel–Moore homology $H_i^\text{BM}(X)$ such that it can be killed by a prime $p$. Under what conditions one can say ...
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### Analogue of Grauert's upper semi-continuity for Bott–Chern cohomology

In Coherent analytic sheaves, one has the following theorem due to Grauert: Let $f: X \rightarrow Y$ be a holomorphic family of compact complex manifolds with connected complex manifolds $X, Y$ and $V$...
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### Geometrically rational variety over a finite field

Let $k=\mathbb{F}_q$ be a finite field, and let $X$ be a smooth projective variety over $k$. Suppose that $X_{\overline{k}}$ is birational to $\mathbb{P}^n_{\overline{k}}$, do we know (1)If $X$ is ...
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### Tannakian fundamental group of automorphic representations

Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$. If this is a Tannakian category, it has an associated ...
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### First cohomology of tangent sheaf of rational curve

Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$. Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...
### Number of independent conditions imposed by points in $\mathbb P^3$ on $\mathcal O_{\mathbb P^3}(3)$
Let $Z$ be a zero dimensional subscheme of atleast $10$ distinct points in $\mathbb P^3$ (over $\mathbb C$) such that no quadric passes throught it. Assume that $Z$ also satisfies Cayley-Bacharach ...
There are two languages endow the theory of coherent sheaves with a six functor formalism (that I "know" of), one being formulated in $\text{ProCoh}(X)$ by Deligne and the other being \$D(\...