# 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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### 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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### 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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### Algebraic atlas on smooth manifolds

A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition ...
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### Proving algebraicity of compact Riemann surfaces without Chow's theorem

I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
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### Examples or references for this claim about elliptic Calabi-Yau threefolds

In this article (page 2) , the authors say: "it is expected, based on known examples, that Calabi–Yau threefolds of large Picard rank are always elliptically fibered, perhaps after flopping a ...
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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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### 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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### 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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### 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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### Noetherianity assumptions in Hartshorne's book

It seems that noetherian assumptions are not necessary in many results by Hartshorne, in his book "Algebraic Geometry". How much is this true? Could you please give examples?
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### A "boundary map" for the algebraic equivalence relation of cycles

In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want. Let $X$ be ...
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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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### Is there an algebraic curve over Q which is not modular?

Every elliptic curve $E/\mathbf Q$ is modular, in the sense that there exists a nonconstant morphism $X_0(N) \to E$ for some $N$. It is tempting to extend this definition in a naïve way to an ...
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### Characterization of an Abelian surface

I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E_{1,t}$ and $E_{2,t}$, (I don't know what either $t$ runs through!) such that (1), for any i={1，2}, the closed ...
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### What is a moduli space of Calabi-Yau threefolds?

A Calabi-Yau threefold is a compact Kahler threefold which is simply connected and has trivial canonical bundle. So my question is as in the title. What is the moduli space of such objects? I'm ...
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### Radical of an ideal in the polynomial ring with reducible generators

To find the radical of an ideal can be a very complicate task. Considering ideals in the polynomial ring, I am wondering if this task can be simplified in the case the generators of the ideal have the ...
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Let $\pi:X\rightarrow W$ be a morphism of smooth projective varieties over a field $k$ whose generic fiber is a smooth quadric, and let $r$ be the dimension of the fibers of $\pi$. Does there always ...
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### Grothendieck splitting theorem

By a theorem of Grothendieck we know that all holomorphic vector bundles $E$ on $\mathbb{P}^1_{\mathbb{C}}$ split E = \mathcal{O}_{\mathbb{P}^1_{\mathbb{C}}}(d_1)\oplus\dots\oplus \mathcal{O}_{\...
I would like to understand the irreducible components of a projective algebraic set. Given an irreducible and homogeneous polynomial $H(w,x,y)\in \mathbb{C}[w,x,y]$ we define $H_i(w,x_0,x_i):=H(w,x_0,... 0answers 99 views ### Algebraic morphism as nontrivial topological embedding This is a cross post. According to this article, every knot can be realized as the intersection of$2$non-singular algebraic sets in$\Bbb{R}^4$, one of which is a standard$\Bbb{S}^3$. However this ... 0answers 83 views ### Rationality of quadric bundles Let$\pi:X\rightarrow W$be a flat morphism of smooth projective varieties over a field$k$whose generic fiber is a smooth quadric. Assume that$W$is rational and denote by$n$the dimension of$W$... 0answers 105 views ### 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 ... 1answer 254 views ### 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$V$... 2answers 1k views ### Grauert's criteria for ample line bundles In their book "Compact complex surfaces", W.P. Barth, K. Hulek, C.A.M. Peters and A. Van de Ven refer to the following theorem: Let$X$be a compact complex space and$L$a holomorphic line ... 1answer 240 views ### 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 ... 1answer 488 views ### 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 $... 0answers 113 views ### Normal bundle of a Fano threefold as Brill-Noether loci Let$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}$has two ambient ... 0answers 189 views ### Does the first cohomology of the Hodge bundle over the moduli space of curves$M_{g,n}$vanish? I know the Leray spectral sequence relates this to the cohomology of the relative dualizing sheaf, but I don't know anything about the cohomology of either of these sheaves. 3answers 1k views ### The affine Grassmannian and the Bogomolny equations In "Electric-Magnetic Duality and The Geometric Langlands Program", Sections 9 and 10, Kapustin and Witten describe certain convolution varieties in the affine Grassmannian (and more generally, in the ... 0answers 167 views ### Weakening of weak Lefschetz theorem Is there some sort of general condition that implies for a closed immersion of projective complex varieties$i:Z\hookrightarrow X$, the map on the$n$-th homology sends non$p$-divisible elements to ... 1answer 209 views ### Support of torsion in the Borel–Moore homology Given a complex quasi-projective variety$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$. Under what conditions one can say ... 1answer 274 views ### Tannakian fundamental group of automorphic representations Let$\mathcal{C}_{\mathrm{aut}}(G, F)$be the category of automorphic representations of a connected reductive group$G$over a number field$F$. If this is a Tannakian category, it has an associated ... 2answers 504 views ### Why is the generalized flag variety a “variety”? In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set$X$of Borel subalgebras of a semi-simple Lie algebra$\...
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 ...