Questions tagged [ag.algebraic-geometry]

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

Filter by
Sorted by
Tagged with
3
votes
0answers
75 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 ...
0
votes
0answers
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 ...
6
votes
0answers
127 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 ...
2
votes
0answers
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 ...
1
vote
0answers
156 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 ...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
0answers
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-\...
6
votes
0answers
189 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 ...
14
votes
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 ...
9
votes
1answer
408 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 ...
0
votes
0answers
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 ...
1
vote
0answers
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,\...
5
votes
1answer
272 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: ...
1
vote
0answers
122 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 ...
2
votes
0answers
111 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 $S$ ...
2
votes
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 ...
4
votes
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
0answers
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(\...
5
votes
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 ...
1
vote
1answer
161 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 ...
2
votes
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
votes
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
votes
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
votes
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}_{\...
4
votes
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 $W$ ...
1
vote
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,...
2
votes
0answers
104 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 ...
1
vote
0answers
112 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
votes
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.
6
votes
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
votes
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 $ ...
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 ...
6
votes
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 $V$...
6
votes
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 ...
2
votes
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 ...
3
votes
1answer
167 views

Flatness of finitely presented algebras

Let $R$ be a commutative (noetherian, if needed) ring, let $f_1,\ldots,f_r\in R[x_1,\ldots,x_n]$ and $A=R[x_1,…,x_n]/(f_1,\ldots,f_r)$, when is $A$ flat over $R$? I found a nice answer for the case $n=...
3
votes
1answer
148 views

Characterized maximal ideal [closed]

$\DeclareMathOperator\Alg{Alg}$Let $A$ be a commutative associative algebra with $1$ over $\mathbb{C}$. We define $\Alg(A,\mathbb{C}) $ to be the set of $\mathbb{C}$-algebra maps from $A$ to $\mathbb{...
2
votes
1answer
124 views

Should the identity labelled by red line be $\overline{f(Z)}=X$?

The above picture is from Milne's Etale Cohomology. Suppose $A=\Bbb Z, \mathfrak q=(2T+3)$, consider $Z=\operatorname{Spec} \Bbb Z[T]/(2T+3)\to \operatorname{Spec} \Bbb Z[T]\to\operatorname{Spec} \Bbb ...
3
votes
0answers
113 views

Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
2
votes
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-...
3
votes
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 ...
2
votes
0answers
40 views

Approximation of Grassmanian cubatures through random noise

Let $G_{1,d}$ be the $1$-grassmanian in $d$ dimensions, that is the set of linear projections from $\mathbb R^d$ to $\mathbb R$. We can see it as $\mathbb S(\mathbb R^d)$, as any projection can be ...
1
vote
0answers
58 views

Finding multivariate binomials with a common zero

I have a problem for which I have to find binomials over a multivariate polynomial Ring which all have a common zero. Let $\mathbb{F}[x_1,\dots,x_n]$ be some multivariate polynomial ring over some ...
3
votes
0answers
133 views

Homological stability of Chow varieties

Given a connected component $C$ of the degree $d$ Chow variety of $r$ cycles on $C_{d,r}(X)$ ($X$ is smooth projective variety over $\mathbb{C}$), let $C'$ be another connected component of $C_{d',r}(...
5
votes
1answer
202 views

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 ...
2
votes
0answers
170 views

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 ...
7
votes
1answer
369 views

Relation between ProCoh and solid modules

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(\...

1
2 3 4 5
377