Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,607
questions
2
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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)...
3
votes
1
answer
182
views
Are there geometric $\mathbb{G}_a$-quotients with trivial stabilizers, not being principal bundles?
Consider algebraic $\mathbb{C}$-schemes. The group scheme $\mathbb{G}_a$ is the scheme $\mathbb{A}^1$ with the addition. This is not a reductive group. Here I want to know some examples of $\mathbb{G}...
7
votes
1
answer
1k
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Beilinson's height pairing vs. Néron–Tate
In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arithmetic intersection ...
3
votes
1
answer
494
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Normal bundle to a double line in quartic threefold
I work on complex numbers. Let $Q \subset \mathbb P^4$ be a quartic threefold that contains a double line $l$. I want to compute the normal sheaf of $l$ in $Q$, and I seem to find only two possible ...
155
votes
11
answers
19k
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Why are flat morphisms "flat?"
Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...
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 $\Phi(A)=\mathcal{A}_s/\mathcal{A}_s^0$ ...
3
votes
0
answers
317
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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 ...
9
votes
1
answer
313
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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})\...
12
votes
2
answers
506
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Coordinate ring of universal centralizer (BFM space)
In the paper titled Equivariant (K-)homology of affine Grassmannian and Toda lattice, the authors, Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković, derived the coordinate ring of each ...
3
votes
1
answer
758
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Stacks and Maurer-Cartan elements
One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$.
For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
6
votes
1
answer
234
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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 ...
1
vote
0
answers
101
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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 ...
3
votes
1
answer
215
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Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem
Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
16
votes
3
answers
4k
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Contracting divisors to a point
This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...
3
votes
1
answer
113
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Connected components of a spherical subgroup from spherical data?
This question is in a similar spirit to this one by Mikhail Borovoi.
Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
Losev proved that the spherical $...
4
votes
1
answer
202
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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 ...
7
votes
1
answer
370
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Integral refinements of rigid cohomology
Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid ...
11
votes
1
answer
846
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Cohomological bounds for scalar curvature of an extremal Kähler metric
There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...
3
votes
0
answers
161
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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 ...
4
votes
2
answers
380
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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 ...
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 ...
3
votes
1
answer
153
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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 ...
12
votes
1
answer
796
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Where can I find a collection of new errata for EGA?
There is an extensive list of EGA's errata on the books themselves, but my question is whether new errata, that is those found by various mathematicians after the publication, are collected somewhere.
...
5
votes
1
answer
245
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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 ...
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 $...
0
votes
0
answers
122
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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
172
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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-...
1
vote
2
answers
508
views
Finding regions where multi-variate polynomials are positive
Given a constant $k \in \mathbb N$, and a set of $p$ multi-variate polynomials {$P_j:\mathbb N^n\to \mathbb Z$}$\_{j=1...p}$, with $P_j \not\equiv 0$.
Is the following true:
There exists $n$ sets $...
5
votes
1
answer
699
views
Compact quasi-coherent sheaves
Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category.
3
votes
0
answers
155
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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 ...
20
votes
3
answers
665
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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
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 ...
6
votes
0
answers
145
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 ...
2
votes
1
answer
611
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A "boundary map" for the algebraic equivalence relation of cycles
In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want.
Let $X$ be ...
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 $\...
6
votes
1
answer
528
views
Existence and uniqueness of extensions of a finite flat map
Suppose that $S$ is smooth and that $U\subset S$ is a dense open subscheme. Let $X$ be a scheme (not necessarily smooth) and let $f:X\to U$ be a finite flat morphism. I would like to know whether ...
39
votes
1
answer
3k
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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
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 ...
30
votes
4
answers
3k
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Motivation for zeta function of an algebraic variety
If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be
$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$
where $N_m$ is ...
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}...
11
votes
2
answers
2k
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Motivation for birational geometry
I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
3
votes
1
answer
259
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 ...
46
votes
0
answers
2k
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Mikhalkin's tropical schemes versus Durov's tropical schemes
In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
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 ...
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 ...
0
votes
0
answers
78
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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
139
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 $E$ ...
7
votes
3
answers
1k
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Cohomology of the tangent sheaf of $\mathbb{P}(1,2,3)$
Using the exact sequence
$$0\mapsto\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T_{\mathbb{P}^{2}}\mapsto 0$$
it is easy to compute $H^{1}(\mathbb{P}^...
1
vote
0
answers
85
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 ...
2
votes
0
answers
72
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^...