Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,603
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Is every free additive action on the affine space conjugate to a translation?
Is every free action of the additive group $\mathbb{G}_a$ on the affine space $\mathbb{A}^3$ conjugate to a translation?
In characteristic zero, the answer is yes, and is due to Kaliman. [Kaliman, S. &...
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133
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Relationship between $\infty$-categories of ind-coherent sheaves on the base and total space
My question is vaguely as follows, let $G\to E\to X$ be a principal $\infty$-bundle for some group object $G$ in a category of ($\infty$-)stacks. Can we recover the category $\operatorname{IndCoh}(E)$ ...
- 1,294
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129
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Kodaira-Spencer morphism - complete deformations
Let $X$, $T$ be smooth varieties over $\mathbb C$, $X$ projective, and $\mathcal E$ a coherent sheaf on $X\times T$, flat on $T$. Let $t_0\in T$ be a closed point. Suppose that all the sheaves ${\...
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relations between non-negativity of multivariate polynomials and SOS over gradient ideal
We know there is a necessary condition for the non-negativity of multivariate polynomials in the paper "Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational ...
- 59
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154
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Derived pushforward of a Schur functor, and bounded derived categories of Grassmannians
Consider Grassmanianns over fields of characteristic zero.
Let $i : Gr_{k-1,n} \rightarrow Gr_{k,n+1}$ be the `direct sum' map between Grassmannians. By universal property of Grassmannian, this map ...
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107
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Number of conditions imposed by general points
I encountered with a problem when I read the part of Enriques-Babbage Theorem of the book Geometry of Algebraic Curves Vol. I by ACGH. It is stated on page 112-113 that all subsets of $m$ points of a ...
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4
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144
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Universal property of the category of quasicoherent sheaves of a blowup
We know that if $Z \rightarrow X$ is a closed subscheme of X of ideal $\mathcal{I}$, then if $\pi : Bl_Z X \rightarrow X$ is the projection, $\pi^* \mathcal{I}$ is invertible. Does the category of ...
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228
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Is a smooth projective variety over $\mathbb{C}$ dominated by a Ball?
Suppose that $X$ is a smooth projective variety of dimension $d$ over the complex numbers.
Is it true that there is a ball $\Delta_d=\{ z\in \mathbb{C}^d / \lvert z\rvert<1\}$ and a surjective ...
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276
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What is known about the number of elements needed to generate a given ideal in $k[X_1,\dots,X_n]$?
In Algebraic Geometry by J.S. Milne, after he proves Hilbert's Basis Theorem, he makes the following aside:
One may ask how many elements are needed to generate a given a given ideal $\mathfrak a$ in ...
- 191
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132
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Simple Grothendieck-Riemann-Roch computation with relative Todd class
$\DeclareMathOperator\Tot{Tot}\DeclareMathOperator\ch{ch}\DeclareMathOperator\td{td}\DeclareMathOperator\ker{ker}\DeclareMathOperator\rk{rk}$I was wondering if the following is correct: Let $X=\Tot(L)$...
- 139
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216
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A Brauer group of a double covering of a "well-understood" variety
Let $k$ be a field (it is possible to assume that $k = \mathbb{Q}$ or $= \overline{\mathbb{Q}}$) and $X, Y$ nice varieties over $k$.
Let $f \colon Y \to X$ be a finite flat surjective morphism of ...
- 1,352
12
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1
answer
2k
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Who proved the motivic 6-functor formalism?
In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that
when $...
- 625
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426
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Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
- 21
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How to see that Eisenstein series are eigenfunctions of the laplacian?
Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
- 7,161
3
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196
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Tangent space of a GIT quotient of $GL_{N}$
Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...
- 131
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163
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Splitting of de Rham cohomology for singular spaces
I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
- 433
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87
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About the relationship between Cayley-Chow families and well-defined family of proper cycles
I'm studying Chow varieties introduced in Chapter I.3-4 of "Rational curves on algebraic varieties" [Kol96] by János Kollár and also very interested in the "open" Chow variety ...
- 61
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212
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Higher direct images along proper morphisms in the non-Noetherian setting
Let $f : X \to Y$ be a finitely presented proper morphism. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Do the functors $R^i f_* \mathcal{F}$ preserve any of the following properties:
(1) ...
- 3,301
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143
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Blowing up a reduced ideal in a normal variety
If I blow up a reduced ideal sheaf in a normal variety, is the resulting variety normal?
- 11
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119
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Nullstellensatz + Zariski Density?
My algebraic geometry is a little rusty, so sorry if this is really easy. Here is the situation I have:
I have $n+1$ polynomials $p_1(x_1, x_2, \dots x_n, y)$, $p_2(x_1, x_2, \dots x_n, y)$, $\dots$ , ...
- 2,052
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192
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Vakil's Generalization of qcqs Lemma
(This was also simultaneously asked on math stack exchange: https://math.stackexchange.com/questions/4857715/vakils-generalization-of-qcqs-lemma)
In the most recent notes of Vakil, this is problem 15....
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127
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Cohomology of equivariant toric vector bundles using Klyachko's filtration
I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties.
Whereas detailed literature ...
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223
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GIT quotient of a reductive Lie algebra by the maximal torus
Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
- 2,681
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350
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Can an abelian surface be bielliptic
Is an abelian surface containing an elliptic curve a bielliptic surface?
Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then
$A \to A/E$ is an ...
- 91
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343
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Tate's thesis and Riemann-Roch - $\mathrm{GL}_n$ or twisted version?
I recently learned why the Tate's thesis, especially Poisson summation formula, over a function field $F = \mathbb{F}_q(X)$ of a smooth projective curve $X_{/ \mathbb{F}_q}$ implies Riemann-Roch ...
- 1,911
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122
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Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
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110
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Similar to a $d$-twist but over a cubic field
This question could be related to my old and Duality's newer questions.
I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$:
$$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$
For $...
- 913
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351
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Does $\sum_{n \geq 0} a_n x^n=\sum_{n \geq 0} b_nx^n$ imply $a_n=b_n$ for vector-tuple power series?
My reference is Infinite series in p-adic fields by Keith Conrad.
Corollary 5.6. If $f(x)=\sum_{n≥0} a_nx^n$ has a positive radius of convergence in the $p$-adic field $\mathbb Q_p$ then $f$ is ...
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105
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How many isolated points can a degree $d$ planar curve have?
Let $p(x,y)\in\mathbb R[x,y]$ be a bivariate polynomial of degree $d$. What is the maximum possible number of its acnodes (i.e. isolated roots in $\mathbb R^2$ not counting multiplicities)?
A pretty ...
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104
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Identify the universal centralizer of $G$ as moduli space of 'flat connections' on lagrangians in cotangent bundle of $G^{\vee}$
Note that $\mathbb{C}^*$ can be interpreted as the space of flat $\mathbb{C}^*$-connections on the dual of $\mathbb{C}^*$. Our goal is to find a similar construction for $\mathbb{C}$, particularly in ...
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295
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Exact functor in syntomic cohomology
By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site.
Is it also true for a finite flat ...
- 3,432
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1
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95
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Is every Cartesian biaffine plane affine?
This question concerns the (synthetic) geometry of linear spaces.
Definition 1. A linear space is a pair $(P,\mathcal L)$ consisting of a set $P$ whose elements are called points and a family $\...
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148
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Computing basis of $\mathrm{Pic}(\bar{X})$ for a Del Pezzo surface
Say we are given a degree 2 del Pezzo $X$ given by $w^2=Q(x,y,z)$ where $Q(x,y,z)$ is degree 4. We can compute the exceptional lines by computing the 28 bitangent lines of $Q$ and look at the ...
- 31
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Coordinate transformation for 3-dimensional simplicial cone in $\mathbb{R}^3$
Let $k$ be an algebraically closed field and let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$.Let $X$ be the affine toric variety over $k$ associated to the cone $\sigma$, i.e. set $X$...
- 531
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94
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Necessary and/or sufficient condition for invertibility of the gradient of a polynomial of $m$ variables, viewed as a self map of $\mathbb{R}^m?$
I was wondering whether the following is true, and if not, is something known in this direction?
Let $P:\mathbb{R}^m \to \mathbb{R}$ be a degree $r$ polynomial (not necessarily homogeneous) that ...
- 1,407
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209
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"Simple Limit Argument" in Freitag's and Kiehl's Etale Cohomology
I have a question about an argument used in Freitag's and Kiehl's Etale Cohomology and the Weil Conjecture in the proof of:
4.4 Lemma. (p 41) Every sheaf $F$ representable by an étale scheme $U \to X$,...
- 6,000
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Kapustin-Witten branes and the derived moduli stack of Higgs bundles
A lot has been discussed on overflow regarding geometric Langlands and the physics of Kapustin and Witten's groundbreaking paper https://arxiv.org/abs/hep-th/0604151. I would like to add my two cents ...
- 163
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201
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Smooth compactification of complex varieties and uniqueness
Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$.
Here are a few useful ...
- 1,735
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179
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When inverse image presheaf is already a sheaf
Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field ...
- 6,000
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1
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Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
- 11
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185
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Extending abelian schemes and their polarizations from an open subset
Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every ...
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228
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Finite generation of stack cohomology
Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra?
For instance, $\text{H}^*(\text{B}\mathbf{G}...
- 5,505
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172
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Todd class of blow-up
Let $i:X\hookrightarrow Y$ be an embedding of two non-singular projective varieties over $\mathbb{C}$. Consider the blow-up $f:Y' = Bl_XY \to Y$, and the corresponding embedding $j:E\hookrightarrow Y'$...
- 91
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122
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Growth of Betti numbers in moduli spaces of complex stable curves as the number of marked points vary
$\newcommand{\Mgn}{\overline{\mathcal{M}}_{g,n}} \DeclareMathOperator{\nn}{\mathbb{N}} \DeclareMathOperator{\zz}{\mathbb{Z}}$Let $\Mgn$ be the Deligne−Mumford−Knudsen moduli space of stable curves of ...
- 1,596
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1
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230
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Pure varieties which are neither smooth nor projective
Recall that a variety $X$ over a finite field $k$ is said to be pure if the eigenvalues of the Frobenius on $i^{\mathrm{th}}$ etale cohomology of $\overline{X}:=X\otimes_k \overline{k}$ have ...
- 2,681
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58
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Symbolic polyhedron of a monomial ideal
$\DeclareMathOperator\maxAss{maxAss}\DeclareMathOperator\conv{conv}$Let $I$ be a non-zero monomial ideal and $P$ $\subseteq$ $\mathbb R_+ ^ {n+1}$ be its symbolic polyhedron: then
$$
\alpha(P)= \min \{...
- 11
2
votes
1
answer
180
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Perfect complexes of plane nodal cubic curve
Let $C\subset\mathbb{P}^2$ be a plane nodal cubic curve with a unique singular point $O$ at the origin. Then I consider its normalization, denoted by $\widetilde{C}$ and let $\pi:\widetilde{C}\...
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559
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A curious equation on determinant----linear algebra or algebraic geometry?
I recently find a curious and unexplainable(as seems to me) equation on determinant as follows.
$$3\begin{vmatrix}
a_1 & b_1 & c_1 & d_1 \\
a_2 & b_2 & c_2 & d_2 \\
...
- 305
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1k
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Which revolutions in topology and geometry can we expect in the next 20 years? [closed]
In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
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107
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Nice, concrete example of pl-flipping contraction
In a course I'm giving on the MMP, I am discussing the importance of Shokurov's notion of a pl-flipping contraction for showing that flips exist for arbitrary flipping contractions. Does someone have ...
- 2,098