Questions tagged [ag.algebraic-geometry]

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

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81 views

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 ...
5
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1answer
280 views

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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39 views

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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128 views

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 ...
9
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1answer
409 views

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 ...
14
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1answer
275 views

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 ...
6
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190 views

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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0answers
158 views

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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65 views

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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0answers
134 views

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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51 views

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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90 views

Can dimension jumping fiber be irreducible non-reduced?

Let $X,Y$ be complex algebraic manifolds. Let $f\colon X\to Y$ be a proper surjective morphism. Suppose that for some $y\in Y$, the inverse $f^{-1}(y)$ satisfy $\mathrm{dim}f^{-1}(y)>\mathrm{dim}X-\...
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2answers
253 views

Smoothness of orbit of group scheme

Let $G$ be a smooth affine group scheme over a base $S$. $G$ acts on a scheme $X$ over $S$. Let $x$ be an $S$-point in $X$. Then we have an orbit map $G\to X$. I wonder when the image (set-...
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339 views
+50

From coin flips to algebraic functions via pushdown automata

Background We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
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62 views

Extension of short exact sequence on orthogonal Grassmannians

We work over $\mathbb C$. Let $X=OG(k,V)$ be the orthogonal Grassmannian parametrizing the $k$-dimensional subspaces of $V$, isotropic with respect to a non-degenerate bilinear symmetric form $q$. As ...
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95 views

Base change of cohomology when the cohomology is a torsion

Let $(R,m,k)$ be a discrete valuation ring, where $k=R/m$. Let $X\rightarrow \mathrm{Spec}R$ be a projective, integral and flat $R$-scheme. Let $\mathscr F$ be a coherent sheaf such that $H^i(X,\...
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1answer
305 views

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?
2
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1answer
412 views

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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123 views

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 ...
4
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1answer
507 views

Representability of the diagonal morphism of stacks

Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is representable if and only if the sheaf $\mathcal{Isom}(...
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112 views

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 $S$ ...
4
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1answer
139 views

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 ...
16
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2answers
1k views

Missing exposes in SGA 5, and the composition of the SGA's

Over the past couple of years I had to look in SGA for various results, and I can't but marvel at how poorly constructed it is. In SGA1 expose VII "n'existe pas", SGA 1 references higher SGA's, and so ...
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0answers
182 views

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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0answers
103 views

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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1answer
155 views

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 ...
5
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1answer
336 views

Reference request: Generic k3 surface has Picard number 1

I keep running into the statement that "the generic k3 surface has Picard rank 1". For instance the answer of this question (end) and this paper (following Example 1.1) or this paper (proof ...
2
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52 views

Some questions about purely non-symplectic automorphisms of K3 surfaces and eigenspaces

I am reading this paper. Let $S$ be a (algebraic) K3 surface, an automorphism $\alpha_S\in \text{Aut}(S)$ of finite order $n:= |\alpha_S|$ is purely non-symplectic (of order n) if $\alpha^*_S(\...
26
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3answers
2k views

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 ...
1
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1answer
162 views

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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0answers
152 views

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 ...
2
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0answers
137 views

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 ...
4
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1answer
90 views

Embedding quadric bundles

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 ...
5
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0answers
168 views

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}_{\...
1
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1answer
223 views

Irreducible components of a projective variety

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,...
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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 ...
4
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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 $W$ ...
2
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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 ...
6
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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 $V$...
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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 ...
6
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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 ...
10
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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 $ ...
2
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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 ...
6
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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.
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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 ...
3
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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 ...
4
votes
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 ...
2
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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 ...
6
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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 $\...
6
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1answer
170 views

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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