Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
23,049 questions
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How to prove $D^b(\operatorname{Coh}_Z(X))$ is C-Y for complexes supports on zero section of canonical bundle of a Fano
For a smooth Fano variety $Z$, let $X$ be the total space of its canonical bundle. Let $\operatorname{Coh}_Z(X)$ be the category of coherent sheaves that support on $Z$ set-theorically. How to show ...
- 11
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Does every central extension by $\mathbb{G}_m$ arise from a projective representation?
Let $G$ be a smooth connected linear algebraic group over an algebraically closed field (of characteristic zero if you wish). Let's consider (necessarily central) extensions of $G$ by the ...
- 323
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Universal properties of equivariant perverse sheaves
Let $G$ be a (smooth, connected for convenience) algebraic group acting on a variety $X$ (over a reasonable base scheme). The theory of equivariant perverse sheaves is a functor $\operatorname{Perv}_G(...
- 1,255
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Family of vector bundles over a relative A^1
Let $A=\mathbb{C}[[s]]$, I am looking for an example of a non-trivial vector bundle (or $\operatorname{SL}_n$-torsor) on $\mathbb{P}^{1}_{A[t,t^{-1}]}$ that is trivial over $\mathbb{P}^{1}_{\mathbb{C}...
- 3,552
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Mumford-Knudsen expansion for higher Chern classes
Let $f\colon X \to B$ be a family of varieties over the complex numbers. Assume that $X$ and $B$ are both smooth and projective, and $f$ is faithfully flat. Let $L$ be an $f$-ample divisor on $X$.
The ...
- 2,336
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Stalkwise isomorphic etale sheaves implies locally isomorphism
I was reading the proof of Milne étale cohomology on Brauer groups IV.2.1. Prove that if $A\otimes k(x)$ is a central simple algebra over $k(x)$ for all $x$. Then there exists (finite) etale covering $...
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The analytic Satake equivalence for groups defined over subfields of $\mathbb{C}$
I am starting to learn about the geometric Satake equivalence. In the statement of its $\ell$-adic version: let $G$ be a split reductive group over a field $k$, then the dual group of the Satake ...
- 1,255
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Generating category without taking summands
A well known notion of generating a category $\mathcal{C}$ using a set of objects $S$ is taking direct sums, shifts, summands, and cones between objects in $S$. Let $\mathcal{C} = D^b(X)$ where $X$ is ...
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Is it true that $\operatorname{Alb}(X \times Y ) \cong \operatorname{Alb}(X) \times \operatorname{Alb}(Y) $?
$\DeclareMathOperator\Alb{Alb}\DeclareMathOperator\Pic{Pic}$Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$. My question is, is it true that,
$$\Alb(X \times Y ) \cong \Alb(X) \...
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Reference-request in deformation theory
Let $X$ be a smooth projective variety, and $Z\subset X$ a smooth closed subvariety of $X$. The first order deformations of $Z$ in $X$ are parameterized by $H^0(Z, N_{Z/X})$, while the first order ...
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Question about common zeros of hypersurfaces in projective space
Let $V$ be a vector space of homogeneous polynomials of degree $d$ in $n$ variables, or equivalently a subspace of the linear system of degree $d$ hypersurfaces in projective space $\mathbb{P}^{n-1}(\...
- 23.7k
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MMP over finite fields. State of the art
I would like to know how much of the results on the Minimal Model Programme (MMP) over fields of finite characteristicb which are usually only stated for varieties over algebraically closed fields, ...
- 2,235
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Generic subtorus orbits in a toric variety
Let $X\subset \mathbb{P}^m$ be a projective $n$-dimensional toric variety, with algebraic torus $T$ sitting inside it. This embedding corresponds to a polytope $P\subset \mathbb{Z}^n$, with $m+1$ ...
- 731
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Higher infinitesimal neighbourhoods as symmetric powers, geometrically
Let $k$ be a field of characteristic $0$.
Consider the $k$-algebra $A = k [t_1, \ldots, t_n] / ({t_1}^2, \ldots, {t_n}^2)$.
We can think of $A$ as being the $k$-algebra generated by $n$ linearly ...
- 17.5k
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Real structure of dual vector bundle
Suppose $V$ is a holomorphic vector bundle on a Riemann surface X which is endowed with an anti-holomorphic involution $\sigma_X: X\longrightarrow X$. $V$ is said to have real structure if $\exists$ ...
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The Intersection number on a complex noncompact variety
Let $X$ be a variety over $\mathbb{C}$ and $C \subset X$ be a proper curve. Let $L$ be a line bundle on $X$. Then, the intersection number $L \cdot C$ is given by $\deg L|_{C^\nu}$, where $C^\nu$ is ...
- 573
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Rank of tensor product of irreducible representations over finite symmetric group
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of equivalence classes of irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $\...
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Properties of orbit under group scheme action & dimension count formula
Let $X/k$ be a scheme over base field $k$ and $G$ an group scheme (also over $k$) acting on $X$ (ie there is an algebraic map $\sigma: G \times X \to X$ satisfying some compatibility conditions).
Let $...
- 3,842
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Meaning of subscript $h$ in in Hochschild Homology
This is from Hesselholt's On the de Rham-Witt complex in mixed characteristic. In the course of the proof of Thm. 6.2.4 the author uses the notation
\begin{equation*}
{}_{h}\operatorname{TR}_{q}^{...
- 1,953
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Reference for Clifford algebras and orthogonal grassmanians
I'm reading about spinor bundles in Kuznetsov's paper https://arxiv.org/pdf/math/0512013. On page 17 he states that the orthogonal Grassmanian $\mathsf{OGr}(m,2m)$ with respect to a non-degenerate ...
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Does coarse moduli morphism induce isomorphisms on rational $\mathbb{A}^1$ homology sheaves?
Let $S$ be a Noetherian scheme and $\mathrm{Sm}_S$ the category of smooth schemes over $S$ of finite type. A motivic space over $S$ is an $\mathbb{A}^1$-local Nisnevich $\infty$-sheaf on $\mathrm{Sm}...
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Determinantal elimination for $f_i=x_i(y+t_i)-1$: is there an analogue for $f_i=x_i(y+t_i z+s_i)-1$?
Consider the polynomials
$$
f_i = x_i (y + t_i) - 1,
$$
where the variables are $x_i$ and $y$.
Experiments indicate that, if the coefficients $t_i$ are algebraically independent, then there exists a ...
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Set theoretic complete intersection property of an algebraic curve
I am working on a research problem and would like clarification on the following question.
Let $V$ be an irreducible algebraic curve in $\mathbb{C}^d$ with $d>3$. Suppose $V$ is contained in an ...
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270
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Prismatic fundamental group
Prismatic cohomology is a cohomology theory introduced by Bhatt and Scholze around 2019.
It provides a unified framework for p-adic Hodge theory, bringing together various cohomology theories such as ...
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Curvature characterization of Kodaira–Iitaka dimension
Let $X$ be an $n$-dimensional complex manifold and let $L$ be a holomorphic line bundle over $X$. We denote by $\kappa(L):=\kappa(X,L)$ the Kodaira–Iitaka dimension of $L$.
In the extreme case, say, $...
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