Questions tagged [ag.algebraic-geometry]

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

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A question related to the strong Oda conjecture

A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining ...
Hugh Thomas's user avatar
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3 votes
1 answer
261 views

Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?

Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the ...
klerk's user avatar
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0 votes
1 answer
127 views

Double disk bundle

Double disk bundle: A smooth, closed manifold $M \cong DB^{-} \cup_L DB^{+}$ where · $B^{±}, L$ smooth, closed manifolds · $D^{l± +1} → DB^{±} → B^{±}$ smooth disk bundles such that $S^{l±} → L \cong ...
Dimpi Paul's user avatar
2 votes
0 answers
62 views

Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles

Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...
Matthias's user avatar
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2 votes
0 answers
137 views

Vanishing differential of Brown-Gersten-Quillen spectral sequence

Let $k$ be an algebraically closed field of characteristic zero and $X$ be an affine, simplicial toric 3-fold over the field $k$. I am trying to use the Brown-Gersten-Quillen spectral sequence to ...
Boris's user avatar
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2 votes
0 answers
75 views

Simply connectedness of leaves of a foliation on an complex manifold

Now I'm searching about leaves of foliation in the following special setting. Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...
George's user avatar
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2 votes
1 answer
152 views

Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$

It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...
Partha's user avatar
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1 vote
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125 views

Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form

Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
kvicente's user avatar
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2 votes
1 answer
197 views

Surjective étale map from simply connected curve over $\mathbb{C}$

Let $X$ be a simply connected algebraic curve over $\mathbb{C}$ and $f:X\rightarrow \mathbb{A}^1_{\mathbb{C}}$ is a surjective étale map. Then is it true $f$ is finite? All the domains of non finite ...
George's user avatar
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+50

Is there a correction to the failure of geometric morphisms to preserve internal homs?

Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
Cameron's user avatar
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1 answer
189 views

Fano scheme of cubic threefold in characteristic 3

Altman and Kleiman in Foundations of the theory of Fano schemes prove that if $F = F_1(X) \subset \mathbb G(1, 4) = G$ is a Fano scheme of lines on cubic threefold in $\mathbb P^4$ with at most ...
Alexander Golys's user avatar
3 votes
1 answer
145 views

Pushforward of exceptional vector bundle is spherical for local P^2

I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical ...
cdsb's user avatar
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1 vote
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114 views

What can we say when a module of differential is free?

Let $\mathbb{C}$ complex number. $R=\mathbb{C}[x,y]/(f)(f\in \mathbb{C}[x,y])$ If the module of differential $\Omega_{R/\mathbb{C}}$ is free $R$ module of rank one, what can we say about $R$. How far ...
George's user avatar
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5 votes
0 answers
185 views

Applications of Langlands for GLn explicit reciprocity laws other than elliptic curves

Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$ for $n\ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves? I've never found such ...
Cloudifold's user avatar
1 vote
0 answers
113 views

About the proof that deformation of canonical singularity is still canonical singularity

I was reading Professor Kawamata's paper Deformation of Canonical Singularity (which also appears in the book Algebraic Varieties: Minimal Models and Finite Generation Corollary 2.12.11) the main ...
yi li's user avatar
  • 183
4 votes
0 answers
283 views

Merits of derived geometry

What are the merits of derived geometry? More precisely, which specific mathematical problems that can be formulated without this machinery have been solved using it? If those problems exist, could ...
HCH's user avatar
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16 votes
0 answers
300 views

Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )

Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
Alexander Chervov's user avatar
4 votes
1 answer
171 views

Irreducible components of rigid varieties

I'm reading IRREDUCIBLE COMPONENTS OF RIGID SPACES (by Conrad). In this paper he defines the irreducible component of a rigid variety $X$ to be reduced image of a connected component of $\tilde X$ (...
Richard's user avatar
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3 votes
1 answer
133 views

On a multiplication map of divisorial sheaf

It maybe a very foolish question, but I can't prove or disprove this. Suppose $X$ is a normal variety, and $L$ a $\mathbb{Q}$-line bundle on $X$. I.e., there is a $\mathbb{Q}$-Cartier Weil divisor $D$ ...
nariri's user avatar
  • 350
2 votes
1 answer
130 views

Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane

Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral: $$I(v) := \int_0^1 \det(v(t),v'(t))dt$$ tells us about $v$, where $\det(v(t)...
stupid_question_bot's user avatar
2 votes
0 answers
143 views

On the definition of the relative canonical divisor

Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...
Don's user avatar
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3 votes
0 answers
88 views

Vanishing cycles and component groups

Let $A$ be an abelian variety over a local field $K$ and assume it has toric reduction. Then two classical invariants associated to this are the component group $\Phi(A)=\mathcal{A}_s/\mathcal{A}_s^0$ ...
curious math guy's user avatar
6 votes
1 answer
234 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
319 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
  • 83
4 votes
1 answer
227 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
101 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
  • 183
9 votes
1 answer
313 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
  • 350
3 votes
0 answers
161 views

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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0 votes
0 answers
160 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
249 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
  • 497
0 votes
0 answers
122 views

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
  • 107
1 vote
0 answers
173 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
155 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
3 votes
0 answers
92 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
2 votes
1 answer
175 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
  • 2,863
4 votes
1 answer
202 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
162 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
147 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
669 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
  • 1,187
3 votes
1 answer
153 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
126 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
39 votes
1 answer
3k views

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
292 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
203 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
  • 1,651
4 votes
2 answers
380 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
260 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
  • 5,099
6 votes
1 answer
395 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
  • 523
1 vote
0 answers
93 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
78 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
141 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
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