Questions tagged [ag.algebraic-geometry]

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

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Calculating vertical displacement to intercept diameter of circle after horizontal shift [closed]

I can’t come upon a solution for the question below. The background of the actual work is not interesting for the question, so I won’t explain why I need the solution. I would be very thankful if ...
Darius E.'s user avatar
6 votes
1 answer
177 views

Criteria for when Gauss-Manin sheaves are vector bundles

Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
Aitor Iribar Lopez's user avatar
3 votes
0 answers
142 views

The local global principle for differential equations

Are there any good reference to tackle the problem below? Or, are there any know result? Problem Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...
George's user avatar
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-3 votes
0 answers
84 views

On unitary manifold [closed]

For integer $d$, we can define unitary matrices $\mathbb{U}:=\{U\mid U^*U=UU^*=I_d\}$. This forms a manifold of dimension $d^2$. Suppose we are given a submanifold of $\mathbb{U}$, denoted as $\mathbb{...
gondolf's user avatar
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3 votes
1 answer
114 views

Comparing Kummer maps to étale homotopy at finite level

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Hom{Hom}\newcommand{\et}{\mathrm{et}}\newcommand{\top}{\mathrm{top}}$In Voevodsky's paper "Étale topologies of schemes over fields of finite ...
Thigh High Crocs's user avatar
1 vote
0 answers
79 views

Extension of MMP from the central fiber to some neighborhood

I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 ) There is a theorem about the extension of MMP step when the central fiber has ...
yi li's user avatar
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7 votes
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153 views

Maybe a folklore natural map between reflexive pullbacks

In the introduction of [HK04], it is proposed that for a morphism between varieties $f:X'\to X$, and a coherent sheaf $\mathcal{F}$ on $X$, there is a natural map $\alpha:f^*(\mathcal{F}^{\vee\vee})\...
nariri's user avatar
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3 votes
0 answers
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A relative Abel-Jacobi map on cycle classes

I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations. Background: Suppose $X$ is a smooth projective ...
Asvin's user avatar
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137 views

Why $k((x,t))$ can not be a local field?

If $k$ is a finite field, then $k((x))$ is a local field, and we can define a discrete valuation on $k((x))$ with respect to which it is complete. It is sometimes called a 1-dimensional local field. I ...
MAS's user avatar
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5 votes
1 answer
214 views

Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
a_g's user avatar
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0 answers
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Projective subvarieties are closed?

I want to show that projective subvarieties of a quasi-projective variety are closed. One possible solution should be the following: Let $W \subseteq \mathbb{P}^n$ be a quasi-projective variety and $V ...
psl2Z's user avatar
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1 vote
0 answers
134 views

Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$. Algebraically its $Spec$ is quite different from $k$. For example: it has plenty non-trivial "line-...
Alexander Chervov's user avatar
3 votes
0 answers
134 views

What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?

Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
Alexander Chervov's user avatar
2 votes
0 answers
80 views

Existence of extensions of a flat projective morphism

Suppose that $S$ is a noetherian integral scheme and $U\subset S$ is an open subscheme. Let $f:X\to U$ be a flat projective morphism. I would like to know whether (or when) $f$ can be extended to a ...
Nachhauseweg's user avatar
1 vote
0 answers
104 views

Noetherian local ring with non-lci formal fibers

I am looking for an example of a noetherian local ring $A$ such that its formal fibers are not complete intersection rings in the sense of https://stacks.math.columbia.edu/tag/09PY (i.e., there is a ...
gdb's user avatar
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4 votes
1 answer
174 views

Equivariant projective embeddings with optimal dimension

Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find ...
Misha Verbitsky's user avatar
4 votes
1 answer
150 views

Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?

I am currently reading Katz and Mazur’s Arithmetic moduli of elliptic curves and I am puzzled by a statement in the discussion of the $[\Gamma_0(N)]$ moduli problem in Chapter 3. The authors define a $...
Aphelli's user avatar
  • 330
6 votes
0 answers
139 views

$K_0$ of arithmetic surfaces

In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated ...
Daniel Schäppi's user avatar
20 votes
3 answers
551 views

Examples when quantum $q$ equals to arithmetic $q$

First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor. In the world of quantum mathematics, the letter $q$ is a standard ...
Estwald's user avatar
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3 votes
1 answer
138 views

When is a del Pezzo surface a conic bundle?

I am considering over a field $k$ which is not algebraically closed, characteristic 0, and perhaps contains all the complex roots of unity that may appear. Feel free to realize it as some function ...
fp1's user avatar
  • 71
2 votes
0 answers
116 views

Classifying stack for finite flat group scheme

Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...
mhahthhh's user avatar
  • 381
35 votes
1 answer
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Clausen–Scholze's Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now?

In the recent lecture series run jointly from IHÉS and Bonn, Clausen and Scholze have reworked—again—their foundations of geometry to focus attention on not arbitrary condensed sets and solid modules ...
David Roberts's user avatar
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3 votes
1 answer
282 views

Does the absolute Frobenius induce the identity on étale topoi?

Let $X$ be a scheme defined over $\mathbb{F}_p$ and denote by $X_{et}$ its étale topos . Associated to $X,$ we can consider the absolute Frobenius map $F_X: X \rightarrow X$ which gives an associated ...
etalemorphisms's user avatar
3 votes
1 answer
182 views

Etale cohomology of relative elliptic curve

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme. Let $R^1f_*\mathbb{Q}...
kindasorta's user avatar
  • 1,473
4 votes
2 answers
364 views

“Geometric” vs Homotopical completion

There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them. The first one is the “homotopical” (or maybe it should be called ...
Grisha Taroyan's user avatar
3 votes
1 answer
241 views

Is there a variety which is not locally set theoretic complete intersection?

A variety $X$ is a locally set theoretic complete intersection in a nonsingular variety $Y \supseteq X$ if each point in $X$ has a neighborhood $U$ in $Y$ such that $X \cap U$ is set theoretically the ...
pinaki's user avatar
  • 5,064
6 votes
1 answer
382 views

Reference request: good reduction equivalent to crystalline étale cohomology

Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable ...
Richard's user avatar
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1 vote
0 answers
85 views

One question about Manetti surface

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof. Theorem 5.2 states that fixed a ...
RedLH's user avatar
  • 31
0 votes
0 answers
73 views

Projectivity of equivariant K-theory of toric variety

I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups. In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
onefishtwofish's user avatar
2 votes
0 answers
132 views

About pushforward of a sheaf of divisor

Let $X$ be a normal variety over an algebraically closed field of arbitrary characteristic, $f:X'\to X$ a log resolution, $L$ a Cartier divisor on $X$, and suppose $L\sim_{\mathbb{Q},f}E$, where $E$ ...
nariri's user avatar
  • 296
1 vote
0 answers
82 views

Mirror of a local K3 surface

Is there any description of a mirror manifold of a (non-compact) Calabi-Yau threefold given by the total space of the trivial line bundle on a K3 surface? If yes, in what way is it a mirror? Thanks ...
Cranium Clamp's user avatar
2 votes
0 answers
66 views

Projective resolution of a quiver with relations

How do we compute the projective resolution of a representation of a quiver with relations. For example consider the Beilinson quiver $B_4$ $. with the relations ­$\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
user52991's user avatar
  • 159
2 votes
0 answers
208 views

What are the Hodge and log Hodge groups of $M_{g,n}$?

I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
Leo Herr's user avatar
  • 1,084
2 votes
1 answer
164 views

Lifting of quadrics containing a curve

Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
user267839's user avatar
  • 5,948
1 vote
1 answer
186 views

Shrinking the base field of an affine variety

This is a question on algebraic geometry/commutative algebra. Let $K,L$ be fields of characteristics zero and let $K\subset L$ be a field extension (I am interested in the case when this is ...
S.J.'s user avatar
  • 21
2 votes
0 answers
110 views

How to find a single-variable polynomial in a zero-dimensional ideal?

Given finitely many multivariate polynomials with algebraic coefficients that generate a zero-dimensional ideal, is there an easy way to find a nonzero single-variable polynomial in this ideal? If we ...
Dustin G. Mixon's user avatar
2 votes
1 answer
164 views

Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)

I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia. Let $f:X = \text{...
user267839's user avatar
  • 5,948
1 vote
0 answers
48 views

Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians

I've came across this problem while thinking about some properties of fat schemes. Let me give you an explicit (motivating) example: We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
gigi's user avatar
  • 1,333
1 vote
0 answers
78 views

étale, smooth, and unramified for product of schemes

The question comes from Liu's book. I already asked it on mathstack and I post here since I didn't get any answer at now (https://math.stackexchange.com/questions/4873827/%c3%a9tale-smooth-and-...
Analyse300's user avatar
6 votes
1 answer
300 views

Is there an English translation of Monique Hakim's thesis?

Monique Hakim's thesis, published in 1972 as Topos annelés et schémas relatifs, has been referenced on a multitude of occasions. But I struggle to find a translation into English, even an informal one....
xuq01's user avatar
  • 1,054
3 votes
1 answer
491 views

Why is this polynomial factorizable? [closed]

I met a curious problem on factorizing a homogenerous polynomial of degree 9. Problem: Show that the following polynomial can be divided by $(a_1+a_2+a_3)$: \begin{align} &\quad\left| \begin{array}...
LichenSDU's user avatar
  • 305
11 votes
1 answer
883 views

Relation between motives and geometric Langlands

When working over a number field (or a function field over a finite field), one predicts that the Langlands program is related to the theory of motives over this field. There are several ways I have ...
JustLikeNumberTheory's user avatar
2 votes
1 answer
351 views

Zero set of prime ideal

Let $k$ be a field of characteristic $0,$ but not necessarily algebraically closed. Let $P$ be a prime ideal of $k[X]$ (where $X$ is an $n$-tuple of variables), and $V := \{x \in K^n: \forall p \in P\ ...
Antongiulio Fornasiero's user avatar
2 votes
1 answer
96 views

Normality and integrality of schemes and splitting of map from structure sheaf to (derived)pushforward of structure sheaf along proper birational map

Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, ...
Snake Eyes's user avatar
2 votes
0 answers
101 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
3 votes
1 answer
268 views

Riemann-Hilbert problem via quiver description

The moduli space of Fuchsian systems over $\mathbb{P}^1$ with prescribed adjoint orbits conditions at poles a.k.a. additive Deligne-Simpson problem can be presented under purely quiver description.The ...
TaiatLyu's user avatar
  • 111
2 votes
0 answers
108 views

Some questions about $\ell$-adic monodromy

I'm stucking on the proof of the Lemma 3.12 of A p-adic analogue of Borel’s theorem. Here $\mathcal A_{g,\mathrm K}$ is just a shimura variety defined over $\mathbb Z_p$, and full level $\ell$ ...
Richard's user avatar
  • 481
4 votes
0 answers
219 views

When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?

Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$. It is known that for two elements the following result holds: $\langle u,v \rangle$ is a maximal ...
user237522's user avatar
  • 2,783
4 votes
1 answer
303 views

Residues and blow ups

On a 2-dimensional complex manifold consider two functions which are meromorphic with singularities along two divisors which meet at a point. There is a residue from these meromorphic functions (...
Edwin Beggs's user avatar
  • 1,213
2 votes
1 answer
149 views

One question about K-moduli space of smooth plane conic curves

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" example 4.5 (2) (b) ADL 19 and l have some confusions. From Li-Sun's paper "Conical Kähler-...
RedLH's user avatar
  • 31

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