Questions tagged [ag.algebraic-geometry]

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

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2answers
346 views

Reference request: the geometry of vanishing cycle

I’m currently studying basics on étale cohomology, by Fu’s and Milne’s book. The formalization of vanishing cycle and nearby cycle particularly interests me. I realized it may relate with reduction ...
8
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1answer
394 views

Any news about equivalences of periodic triangulated or $\infty$-categories?

There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted ...
1
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0answers
139 views

Which compact complex surfaces have positive Euler characteristic?

Let $X$ be a compact complex surface. The classification of such surfaces is well-known, bearing the name of Enriques and Kodaira. The classic book, which likely answers my questions here, is the book ...
1
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1answer
190 views

Singularities of Chow varieties

Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C_{d,r}(X)$. The Chow variety can have many topologically connected ...
3
votes
1answer
255 views

Relation between the cohomology group of a curve and the cohomology group of its jacobian

Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that ...
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0answers
87 views

Direct sum of localizations of a module

Given a module $M$ over a local Noetherian ring $R$, when can I assume that $$M =\bigoplus_{\mathfrak{p} \in \text{Supp}(M)} M_{\mathfrak{p}}$$? I suspect that something like Cohen-Macaulay would do ...
3
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0answers
140 views

Finite generation of algebraic $K$-theory with finite coefficients

Given a smooth connected complex quasi-projective variety $X$, is it possible that $K_i(X, \mathbb{Z}/l)$ to be infinitely generated for $i>0$? I think Quillen-Lichtenbaum implies that above the ...
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0answers
54 views

Algebraic sets $A\subseteq F^{n}$ of prime characteristic where $|A\cap L^{n}|=|L|^{n-r}$ for almost all finite subfields $L\subseteq F$

Let $p$ be a prime. Let $K$ be the algebraic closure of a finite field characteristic $p$. Let $f:K^{n}\rightarrow K^{r}$ be a polynomial function. Let $A=\{\mathbf{x}\in K^{n}\mid f(\mathbf{x})=0\}$. ...
6
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1answer
211 views

Why is the scheme of isomorphisms of sheaves affine over the base?

Suppose $S$ a noetherian base scheme, $X \to S$ is projective and $F, G$ are coherent $\mathcal O_X$-modules. Then by EGA (7.7.8) and (7.7.9) there exists a scheme $H = \underline{\operatorname{Hom}}...
12
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1answer
377 views

Katz's $\ell$-adic Airy sheaf

The Airy differential equation $$y''(x)\ = \ xy(x)$$ is one of the simplest irregular differential equations (so not determined by its monodromy data, there is more structure, the Stokes data). ...
5
votes
1answer
200 views

$y^3 = x^4 + x + 2$, and rational points on rank 0 Picard curves

Do there exists rational numbers $x$ and $y$ such that $$ y^3 = x^4 + x + 2 ? $$ Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
5
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1answer
179 views

On realizing a topos of sheaves as a topos of equivariant sheaves

This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\...
2
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0answers
327 views

Relative homology in Fargues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they ...
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0answers
86 views

Galois extension of $ C_{k} $ field and irreducible polynomial over $ C_{k} $ field

A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...
7
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1answer
279 views

Gorenstein varieties: why the two definitions are equivalent?

There are two definitions of Gorenstein singularities in the literature. Using Grothendieck's (or Serre's) duality, one defines the "dualizing sheaf" an object $\hat K_M$ of derived category ...
1
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1answer
76 views

Conics on the complete intersection of two quadrics

Let $Q_1, Q_2\subset \mathbb{P}^4$ be two degree two smooth hypersurfaces, and $S:=Q_1\cap Q_2$ be their complete intersection. If $S$ is smooth, then it is known that $S$ is a del Pezzo surface of ...
2
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0answers
72 views

Compact Kähler manifold with $\kappa(X)=0$ but $K_X$ is not holomorphically torsion

Let $X$ be a compact Kähler manifold. If $K_X$ is semi-ample, then the vanishing of the Kodaira dimension $\kappa(X)=0$ is equivalent to $K_X$ being homomorphically torsion. Does someone have an ...
2
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0answers
82 views

Hilbert-function-like results for weighted projective spaces

Let $R = k[X_1,\ldots,X_N]/I$ be a finitely generated graded generated $k$-algebra, where the $X_i$'s have nonequal degrees, say $\deg(X_i)=a_i$, and $I$ is a (weighted-)homogeneous ideal. We can ...
3
votes
1answer
244 views

Is an equivariant projective morphism equivariantly-projective?

Let everything be over $\mathbb{C}$. Consider two varieties $X,$ $Y,$ where $X$ is normal and $Y$ is affine, having regular $\mathbb{C}^*$-actions and a $\mathbb{C}^*$-equivariant projective morphism $...
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0answers
140 views

Flatness of fixed points scheme

Let $f:X\to Y$ be a flat morphism of algebraic varieties (let's say over $\mathbb C$). You can assume that $Y$ is smooth and $X$ is normal. Assume that $X$ has a fiberwise action of the multiplicative ...
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0answers
66 views

When every principal annihilator is prime

Let $ M$ be a unit $ R$-module over commutative ring $R.$ Is there any equivalent condition that for every element $x $ of $M $ either $ann (x)=R$ or $ann (x)$ is a prime ideal of $R$? where $ann (x)=\...
9
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1answer
560 views

When is Bialynicki-Birula decomposition a paving?

Suppose we have a smooth algebraic variety $X$ with an action of $\mathbb{C}^*$ with finitely many fixed points. Suppose $X$ can be covered by invariant quasi-affine open sets and suppose for each $x\...
3
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1answer
166 views

Countably many isomorphism classes of reductive groups over a field with countable Brauer and Witt groups

Assume a field has a countable Brauer group and a countable Witt group. Are there countably many isomorphism classes of reductive groups over it?
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80 views

Quotient $(V -S)/G$ is a quasi-projective variety for every closed $S \subset V$ with free $G$-action

I have a question about the statement of Remark 1.4 following on Thm 1.3 from Burt Totaro's paper "The Chow Ring of a Classifying Space" (p. 4): Let $G$ be a reductive group over a field $k$....
3
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0answers
117 views

Does existance of crepant resolution of tangent space imply existance of crepant resolution globally in the algebraic setting?

Suppose $X$ is smooth proper algebraic $\mathbb C$-variety with algebraic action of a finite abelian group $G$. Suppose I know that $X/G$ (good geometric quotient) exists and it is normal Gorenstein ...
2
votes
1answer
188 views

Galois invariant line bundle and base change

Let $K$ be a number field and consider a finite Galois extension $L|K$. Moreover let $X$ be a projective, regular, integral variety over $K$. After a base change we obtain a morphism of varieties $f:...
2
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0answers
182 views

Has anyone written down an approach to the Lenard-Magri integrability scheme via algebraic geometry?

I’ve been thinking about the algebro-geometric meaning of the Lenard-Magri scheme of getting an integrable system from a pair of compatible Poisson structures. I think one might be able to prove a ...
0
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0answers
137 views

Complex orbifold and blowing up

Let $M$ be a compact complex surface with finite singular points. (i.e. for regular points $x$ there are manifold coordinate charts $\varphi:U_x\to\mathbb C^2$, for singular points $y$ there are ...
2
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0answers
157 views

Integer points on genus 1 curves using CAS

How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.? As a specific example, do ...
12
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0answers
366 views

Have the details in Voevodsky's original approach to Milnor's conjecture been filled in?

In Bloch-Kato conjecture for $\mathbb{Z}/2$-coefficients and algebraic Morava $K$-theories Voevodsky claimed a proof of Milnor's conjecture under some assumptions. Later he came up with a different ...
16
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1answer
496 views

Instances of "correcting" the compact objects of a category?

I sense a familial resemblance between the following situations: Equivariant homotopy theory: The $\infty$-category $Spaces^{BG}$ of Borel $G$-spaces fails to have a compact unit (= terminal object), ...
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0answers
128 views

Representations of elliptic curves over arbitrary fields

An elliptic curve over a field $k$ is a commutative algebraic group, so we can ask what its algebraic representations are, in particular what its characters (1-dimensional representations) are. Have ...
4
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0answers
198 views

An Akbulut cork with a simple equation?

Is there an Akbulut cork that is diffeomorphic to a complex affine surface that can be given by a simple equation? (for example a surface given by zeros of a low-degree polynomial in $\mathbb C^3$) ...
4
votes
0answers
145 views

What is the fundamental group in noncommutative geometry à la Efimov, Kaledin, Vologodsky?

In the study of DG categories one encounters many different homologies. Is there an interesting notion of the fundamental group?
5
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0answers
77 views

Orlik-Solomon algebra and hyperplane complements in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$. Given a ring $R$ ...
2
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0answers
135 views

Conics on Gushel-Mukai fourfold

Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\...
1
vote
1answer
240 views

Does $L$-functions of elliptic curves over $\mathbb{Q}$ being meromorphic obviously imply modularity?

If I somehow know that for each elliptic curve over $\mathbb{Q}$ the $L$-function has a meromorphic continuation to the whole plane can I easily deduce modularity from that? If not is there a way to ...
2
votes
1answer
114 views

When the annihilator of each nonzero submodule is prime

Let $M$ be a fixed faithful $R$-module over integral domain $R$. Is there any equivalent condition (on $R$ or on $M $) under which the annihilator of any nonzero submodule of $M$ to be a prime ideal ...
3
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0answers
121 views

Does the construction of arithmetic toroidal compactification of $A_{g}$ depend on semistable reduction theorem?

If there is a good theory of arithmetic toroidal compactification over $\mathbb{Z}_{p}$ of the Siegel modular variety with deep enough level structure, then it seems like semistable reduction theorem ...
4
votes
1answer
173 views

Quotients and associated graded

$\DeclareMathOperator\gr{gr}$Let $A = \cup_{i=0}^\infty F_i A$ be a filtered commutative ring, $I \subseteq A$ an ideal. Then we have a canonical surjection $$ \gr(A)/\gr(I) \to \gr(A/I).$$ Under what ...
2
votes
1answer
150 views

Open/closed embeddings and the de Rham space

Let $U\to X$ be an open immersion of schemes and denote by $D$ the (say reduced) complement. Then by applying the de Rham functor, we get morphisms $$U_{dR}\to X_{dR}\leftarrow D_{dR}$$ of the ...
4
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0answers
230 views

Morphisms of jacobians

If $X$ and $Y$ are two complex curves then we have $${\rm Pic}(X\times Y) \ \simeq \ {\rm Pic}(X)\times {\rm Pic}(Y)\times {\rm Hom}(J_X,J_Y) \ ,$$ where $J_X$, $J_Y$ are the jacobians of $X$, $Y$. ...
3
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0answers
164 views

An algebraic proof: A line bundle on a curve with a connection must be of degree 0

Let me state it using the language of $D$-modules. Let $X$ be a smooth projective curve and $\mathcal{L}$ a line bundle on it. Assume that $\mathcal{L}$ has a left action of $\mathcal{D}_X$. Then show ...
8
votes
1answer
260 views

Integrating hypercohomology classes

Let $X$ be a complex variety. By Poincare's lemma, its singular cohomology can be computed as hypercohomology of the holomorphic de Rham complex (viewing $X$ as a complex manifold) $$\text{H}^\cdot(X,\...
3
votes
0answers
83 views

Stability of super vector bundles

A super vector bundle is a $\mathbb{Z}_2$-graded bundle, see for example "Heat Kernels and Dirac Operators" of Berline-Vergne-Getzler section 1.3. Does it exist an adapted notion of ...
16
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0answers
423 views

Do $\infty$-categories make Grothendieck duality simpler?

I've heard multiple times that the main difficulty of Grothendieck duality is that triangulated categories don't 'glue well'. In my view, there are 3 parts in understanding Grothendieck duality: We ...
6
votes
2answers
424 views

Open orbits under the action of an algebraic group

Let $k$ be a field, $X$ an algebraic variety, and $G$ a smooth algebraic group, acting on $X$ via $(g,x)\mapsto g\cdot x$. Fixing $x$ in $X$ a $k$-point, there is a map $f_x:G\rightarrow X$ sending $g\...
1
vote
1answer
148 views

Reference for the Hodge diamond of the Iwasawa threefold

Let $X = G/\Gamma$ denote the Iwasawa threefold, where $$G = \left\{\begin{pmatrix} 1 & z_1 & z_3\\ 0 & 1 & z_2\\ 0 & 0 & 1\end{pmatrix} : z_1, z_2, z_3 \in \mathbb{C} \right\},...
1
vote
1answer
75 views

Problem concerning about an $n$-subspace of $ A_{n}(F) $

Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...
2
votes
1answer
248 views

Can you compute the Krull dimension of a subalgebra using ideals?

$\DeclareMathOperator\height{height}$Let $k$ be algebraically closed, $A$ be a $k$-algebra of finite type and $B$ a sub-$k$-algebra of $A$. Assume there exists a largest ideal $I$ of $A$ such that $I \...