# Tagged Questions

**1**

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28 views

### Reconstructing a function from its variants that negate one argument

Call two functions $g(x_1,\ldots,x_n)$ and $h(x_1,\ldots,x_n)$ from complex numbers to complex numbers equivalent if they are the same up to the order of their arguments. Formally: there is a ...

**0**

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92 views

### Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...

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**0**answers

74 views

### Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...

**6**

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**1**answer

214 views

### Is $G_{\operatorname{red}}$ normal in $G$?

Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us ...

**6**

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**1**answer

228 views

### Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...

**4**

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**2**answers

127 views

### Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...

**5**

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**1**answer

198 views

### Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...

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55 views

### canonical bundle skew-symmetric determinantal varieties

By a skew-symmetric determinantal variety (over $\mathbb{C}$), I mean the projectivization of the set of skew-symmetric matrices of fixed size which rank is less than a given integer.
More precisely, ...

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**1**answer

179 views

### Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...

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278 views

### Formal completion of the normal bundle

Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...

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**2**answers

101 views

### Poincaré bundle and Weil pairing for Abelian schemes

In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated.
The same question for the Weil pairing ...

**5**

votes

**1**answer

92 views

### Properties of singularities that are preserved by categorical quotients

Let $G$ be a reductive group acting on an affine singular
variety $X$, and let $X/G$ be the categorical quotient. I know that if
$X$ has rational singularities, then so does $X/G$ ...

**3**

votes

**0**answers

67 views

### Question about a length inequality in algebraic dynamics

Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of ...

**3**

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**1**answer

176 views

### Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...

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votes

**1**answer

236 views

### Algebraic Geometry: Question on terminology [closed]

What is a quadric surface of balanced bi-degree? Do you know of a good reference on this topic?
Thanks in advance for your replies.

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**0**answers

594 views

### The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)

I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor ...

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**2**answers

231 views

### existence and uniqueness of solutions for ODEs in formal power series?

I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference.
Let $k$ be some (algebraically closed, if needed) field. There is a ...

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**0**answers

399 views

### Looking for a paper of Hartshorne

In a famous paper
Hartshorne - Varieties of small codimension,
Hartshorne formulates a conjecture, which roughly says that varieties of small codimension in projective space are complete ...

**5**

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**2**answers

236 views

### Equivariant derived category and invariant divisor

I'm looking for a reference of the following (folklore?) result.
Let $X$ be a smooth projective variety equipped with a $G=\mathbb{Z}/2\mathbb{Z}$ action (we consider the simplest case, everything ...

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**1**answer

185 views

### Quotient of an algebraic group by a closed algebraic subgroup

Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed and normal subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" ...

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**0**answers

135 views

### 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 $P$, ...

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**0**answers

155 views

### How to induce infinitesimal deformations on curves

Let $C_1, C_2$ be two projective curves (a scheme of pure dimension $1$) in $\mathbb{P}^3$.
The Hilbert scheme of curves contains informations of deformations of curves in $\mathbb{P}^3$. The question ...

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**1**answer

172 views

### Relation between Milnor ring and middle dimensional homology of hypersurface

I have suspected that the following is well-known:
If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ...

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**1**answer

314 views

### Basics on anabelian geometry and Grothendieck's section conjecture

Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...

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104 views

### Which one is the correct assertion in the end? (projectivity of moduli of polarized varieties)

I randomly came across a 2007 article by Kollár in which the author makes a mathematical statement and explicitly states that it is in contradiction with a result contained in a 2004 article by ...

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**1**answer

122 views

### Most general “finiteness of de Rham cohomology” statement for holonomic $D$-modules in the algebraic case?

Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) ...

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**1**answer

129 views

### good reference for the Hitchin fibration

can you please recommend me a good reference to learn about the Hitchin fibration in the language of algebraic geometry?

**6**

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166 views

### The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor

As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...

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votes

**1**answer

188 views

### 'Cohomologically approximating' a $\mathbb{Q}[[t]]$-scheme by a one over the henselization of $\mathbb{Q}[t]$?

For certain matters the henselization $R$ of $\mathbb{Q}[t]$ at $0$ is a 'reasonable approximation' for $\mathbb{Q}[[t]]$ (Artin's approximation theorem and so on). Now, I would like to study certain ...

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**1**answer

276 views

### Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem

By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem ...

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356 views

### Kummer's quartic surface and the Dirac operator

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth ...

**3**

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**1**answer

245 views

### When do tamely ramified Belyi maps exist in characteristic p?

Let $p$ be a prime. Let $d$ be a number. Let $\lambda_0,\lambda_1,\lambda_{\infty}$ be three partitions of $d$ whose parts are prime to $p$. Consider the following question:
Does there exist a ...

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123 views

### Finite Heisenberg groups action on cohomology of line bundles

Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...

**2**

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**1**answer

217 views

### The Bialynicki-Birula Stratification of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian ...

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**1**answer

166 views

### Reference for the Natural Ample Line Bundle on the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line ...

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**3**answers

289 views

### (Non-)Existence of curves of low degree on affine and projective varieties

I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...

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**0**answers

138 views

### Projective schemes with a fixed hyperplane section

Let $H$ be a hyperplane in $\mathbb P^n$, and $X \subseteq H$ be a subscheme. Let $CX \subseteq \mathbb P^n$ be the cone on $X$ from a point $p \notin H$.
Let $Hilb_{CX}$ be the Hilbert scheme whose ...

**0**

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**0**answers

114 views

### Fitting ideal sheaves and determinant bundles

I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...

**1**

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**1**answer

137 views

### formally étale morphisms which are also universally closed

A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally ...

**20**

votes

**5**answers

957 views

### Verlinde's formula

"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT.
Depending on...
• which chiral CFT one considers (does one restrict to WZW models, or not?)
...

**6**

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**2**answers

387 views

### Equivariant Stratifications of a Variety

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are ...

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106 views

### Abel-Jacobi map isomorphism galois representations

Let $X/\mathbb{Q}$ be an irreducible smooth projective curve with a $\mathbb{Q}$-rational point $p$. Then there is a map $\phi: X \rightarrow \textrm{Pic}^{0}(X)$ with the property that $q \mapsto ...

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93 views

### algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights)
Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...

**5**

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**1**answer

168 views

### Stratifications and Filtrations of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that ...

**1**

vote

**1**answer

182 views

### Resolution of singularity of polynomials

Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$.
By Hironaka's desingularization theorem, there exists a birational map ...

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**0**answers

143 views

### Nice references for injective model structures and Quillen functors between motivic homotopy categories

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...

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141 views

### Equivalence of Versions of the Affine Grassmannian

Let $G$ be a compact connected semisimple Lie group. The algebro-geometric definition of the affine Grassmannian is the coset space ...

**5**

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**1**answer

297 views

### What does it mean for a Deligne-Mumford stack to have trivial generic stabilizers?

I have stumbled upon some literature on Deligne-Mumford stacks, and it seems to me, at least superficially, that there is a strong link between DM-stacks which have "trivial generic stabilizers" and ...

**5**

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**2**answers

262 views

### Strata of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...

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**1**answer

153 views

### Group action on a stack and fixed points

This is mostly a reference question. Suppose that I have an action of (say, finite) group $G$ on an algebraic stack $X$ (in my case it is a Deligne-Mumford stack, but this shouldn't matter). As far as ...