Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,545
questions
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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 ...
6
votes
1
answer
177
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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 ...
3
votes
0
answers
142
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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 ...
-3
votes
0
answers
84
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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{...
3
votes
1
answer
114
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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 ...
1
vote
0
answers
79
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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 ...
7
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0
answers
153
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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})\...
3
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0
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142
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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 ...
0
votes
0
answers
137
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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 ...
5
votes
1
answer
214
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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 ...
0
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0
answers
109
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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 ...
1
vote
0
answers
134
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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-...
3
votes
0
answers
134
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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 ...
2
votes
0
answers
80
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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 ...
1
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0
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104
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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 ...
4
votes
1
answer
174
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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 ...
4
votes
1
answer
150
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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 $...
6
votes
0
answers
139
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$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 ...
20
votes
3
answers
551
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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 ...
3
votes
1
answer
138
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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 ...
2
votes
0
answers
116
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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 $\...
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 ...
3
votes
1
answer
282
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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 ...
3
votes
1
answer
182
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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}...
4
votes
2
answers
364
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“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 ...
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 ...
6
votes
1
answer
382
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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 ...
1
vote
0
answers
85
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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 ...
0
votes
0
answers
73
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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 ...
2
votes
0
answers
132
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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 $E$ ...
1
vote
0
answers
82
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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 ...
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^...
2
votes
0
answers
208
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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 ...
2
votes
1
answer
164
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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}...
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 ...
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 ...
2
votes
1
answer
164
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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{...
1
vote
0
answers
48
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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 $\mathbb{P}^2$...
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-...
6
votes
1
answer
300
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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....
3
votes
1
answer
491
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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}...
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 ...
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\ ...
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, ...
2
votes
0
answers
101
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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 ...
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 ...
2
votes
0
answers
108
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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 $\ell$ ...
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 ...
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 (...
2
votes
1
answer
149
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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-...