# Tagged Questions

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

**4**

votes

**1**answer

221 views

### Structure sheaf of affine variety consists of noetherian rings (again)

Let $X\subseteq \mathbb{A}^n$ be an affine variety.
The local ring of $X$ at $p\in X$, given by $\mathcal{O}_{X,p}=\{f\in k(X):f \text{ regular at } p\}$ is noetherian because it is a localization of ...

**4**

votes

**1**answer

163 views

### Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...

**9**

votes

**0**answers

153 views

### Can we just use effective descent morphisms (pure morphisms) as covers?

There are a number of notions of "cover" for a scheme: etale, faithfully flat, fpqc, fppf, Zariski, Nisnevich, etc. Most of these have a nice property, which is that a cover of that type satisfies ...

**6**

votes

**0**answers

77 views

### Center Picard group non-commutative algebra

I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.
Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by ...

**4**

votes

**0**answers

64 views

### Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group

Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...

**5**

votes

**1**answer

142 views

### Is the analytification functor part of a geometric morphism of topoi?

Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally ...

**7**

votes

**0**answers

273 views

### Errata in EGA, collected

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.
...

**3**

votes

**0**answers

79 views

### Irreducibility and flat morphism

Let $Y$ be a quasi projective $k$-variety ($k=\bar k$) and $X\rightarrow Y$ a flat morphism with proper (over $k$) integral $d$-dimensional ($d>0$) fibers over closed points. Can $X$ be reducible?

**8**

votes

**1**answer

245 views

### Finite morphism from a smooth projective curve.

Let $k$ be an algebraically closed field and $C$ be a smooth projective curve over $k$. Let $p$ be a prime number. Does there exists a finite morphism $f : C \to \mathbb{P}^1$ such that the degree of ...

**5**

votes

**1**answer

192 views

### A question regarding lines on a cubic surface

Let $X$ be a smooth cubic surface in $\mathbb{P}^3$. It is a classical theorem of Cayley and Salmon that $X$ contains exactly 27 lines over an algebraically closed field.
In 2002, Heath-Brown proved ...

**4**

votes

**1**answer

208 views

### Intuition behind the definition of finite correspondences

Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over ...

**4**

votes

**1**answer

272 views

### Shafarevich conjecture for abelian varieties

In the paper "Arakelov's theorem for abelian varieties" Faltings proves the Shafarevich conjecture for abelian varieties.
The statement is the following:
Let B be smooth projective a curve, S a ...

**1**

vote

**0**answers

74 views

### Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$

I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when:
(1.) $q=p$
and/or
(2.) $E$ has multiplicative reduction at $q$.
Here, $E$ is an ...

**4**

votes

**1**answer

182 views

### When may “summand of” be dropped from the definition of perfect dg module?

Let $\mathcal{A}$ be a small dg category. In Section 1 of Lunts-Orlov http://arxiv.org/pdf/0908.4187v5.pdf, $Perf(\mathcal{A})$ is defined to be the full DG subcategory of ...

**1**

vote

**0**answers

87 views

### Orders of zeros of section of sheaf

We have a semistable family (fibers has normal crossings and they are reduced, multp 1) $f: X \rightarrow Y,$ of complex curves over a smooth curve $Y.$ The family is smooth over the set ...

**3**

votes

**1**answer

111 views

### Do equivariant morphisms induce representable maps of quotient stacks?

Let $f: X \to Y$ be a $G$-equivariant map between schemes $X$, $Y$ with action of a flat group scheme $G$. Then why is the induced map of algebraic stacks $[X/G] \to [Y/G]$ representable?

**2**

votes

**1**answer

204 views

### Relationship between $ \mathcal{D} $ - modules, Tannakian formalism and Galois theory of monodromy representations

Is there a relationship between $ \mathcal{D} $ - modules, Tannakian formalism and Galois theory of monodromy representations ?
Thanks in advance for your help.

**1**

vote

**0**answers

55 views

### Quadrics passing through a point of a variety that are parametrized by a quadric

Let $X\subset\mathbb{P}^{N}$ be a $n$-dimensional algebraic variety and let $x\in X$. Let us suppose that
$$
\hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\}
$$
...

**1**

vote

**0**answers

69 views

### When is a collection of toric subvarieties in $\mathbb P^r$ generic?

Suppose $Z_1,\dots,Z_n \subset \mathbb P^r$ are toric sub-varieties.
By this I mean, there is a fixed co-ordinate system $\mathbb P^r=\{[z_0:\dots:z_r]\}$ and each subvariety is defined by the ...

**1**

vote

**0**answers

121 views

### Is the category of mixed Hodge modules bi-filtered?

Let $X$ be a smooth complex algebraic variety and let $MHM(X)$ be the category of mixed Hodge modules on $X$, as defined in (Saito, "Mixed Hodge Modules", 1990), (Peters-Steenbrink, "Mixed Hodge ...

**0**

votes

**1**answer

170 views

### Rings with a property similar to integral domains

For an integral domain $R$, the intersection of two non-zero ideals is also non-zero, because the product of any two non-zero elements is non-zero.
Is the converse true, i.e. if $R$ has the ...

**2**

votes

**0**answers

167 views

### Equivariant Riemann-Roch on DM stacks?

Does an equivariant version of (Toen)-Riemann-Roch theorem hold say over a smooth Deligne-Mumford stack $X$ over the complex numbers?
Any references that state this explicitely?
Are there formulas ...

**1**

vote

**1**answer

230 views

### When are Segre- and Veronese embeddings Gorenstein?

Given a Segre product $\mathbb P^m \times \mathbb P^n$, or more generally $\mathbb P^{m_1}\times\cdots\times\mathbb P^{m_n}$, is there a characterization in terms of $m$ and $n$, or the $m_i$, for the ...

**6**

votes

**1**answer

186 views

### Very stable vector bundles

Let $X$ be a smooth curve, and $E$ a rank $r$ vector bundle over $X$, $E$ is said very stable if every nilpotent map $$u:E\rightarrow E\otimes K_X$$ is zero (nilpotent means that the composition ...

**3**

votes

**1**answer

200 views

### Polars of algebraic curves and surfaces

I asked this on Math.StackExchange, but received no response, so trying here ...
A paper I'm reading says the following ...
With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let ...

**1**

vote

**0**answers

87 views

### Weil restriction of fiber products

Let $X,Y,Z$ be smooth geometrically integral proper varieties over a field $K$ where $K/k$ is a finite extension of a number field $k$. Let $R|_{K/k}$ denote the Weil restriction. Suppose we have ...

**8**

votes

**2**answers

435 views

### Moduli space of (all) vector bundles on $\mathbb{P}^1$

It is well known that, by a theorem of Grothendieck, every vector bundle (always assumed coherent in this question; and everything is over the complex numbers) splits as a direct sum of line bundles.
...

**5**

votes

**1**answer

223 views

### Is $MGL$ an $H\mathbb{Z}$-algebra?

Let $\mathrm{MGL}$ be the $\mathbb{P}^1$-ring spectrum over a field $k$ representing algebraic cobordism. Suppose, for simplicity, that $k$ is of characteristic 0. Let $H\mathbb{Z}$ be the motivic ...

**1**

vote

**1**answer

182 views

### Explicit equation for extension of a rational map?

Consider $X = \mathbb{P}^2_k$ and $\mathcal{O}_X(2).$ If we consider the linear system $\mathcal{L}$ of conics passing through the point $[0:0:1]$ we can see that this is spanned by the sections ...

**1**

vote

**1**answer

143 views

### Local nontriviality of genus-one curves over extensions of degree dividing $6^n$

Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}_{p}$ whose degree divides a power of $6$, so that
$C(K)$ is not empty?
(I ...

**1**

vote

**1**answer

640 views

### Problems in some parts of Monique Hakim thesis?

In "Reminiscences of Grothendieck and his school" Luc Illusie says:
"I heard from Deligne that there were problems in some parts. (of Monique Hakim thesis).
Topos annelés et schémas relatifs, ...

**1**

vote

**0**answers

92 views

### Connectedness of fibers of a map related to the secant variety

Let $X\subsetneq\mathbb{P}^{N}$ be a smooth projective variety, and let
$$
S_{X}=\overline{\{(x,y,z)\in X\times X\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\}}.
$$
The secant variety of $X$ ...

**1**

vote

**0**answers

170 views

### Introductions to modern algebraic geometry [duplicate]

I was wondering if people have any feelings on the pros and cons of various introductions to algebraic geometry (among the contenders would be Ravi Vakil's notes, Hartshorne, Mumford, Harris, ...

**0**

votes

**0**answers

146 views

### Chow classes of non-reduced sub-schemes

I am trying to understand the geometric picture of primary ideals, in particular if and how one can define a notion of multiplicity for the underlying geometric 'sets'. My understanding of the subject ...

**7**

votes

**1**answer

300 views

### Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$
$$\require{AMScd} \begin{CD} M^D ...

**8**

votes

**2**answers

316 views

### The class of the diagonal in the symmetric product of a smooth curve

Let $C$ be a smooth curve of genus $g$, and let us consider its $d$-th symmetric product $\textrm{Sym}^d(C)$ and its Jacobian $J(C)$. Fixing a point $p_0 \in C,$ there are two maps $$u_d\colon C_d \to ...

**3**

votes

**1**answer

123 views

### is the normalization of a smooth curve in a tamely ramified finite separable extension of the function field also smooth?

Let $X$ be a smooth proper curve over a field $k$, with function field $K$. Let $L$ be a finite separable tamely ramified extension of $K$, and let $Y$ be the normalization of $X$ in $L$. Is ...

**9**

votes

**0**answers

236 views

### Letters of a bi-rationalist

V.V. Shokurov has written several papers over the course of about 10 years which are called "Letters of a bi-rationalist". Here are the ones that I could find:
Letters of a bi-rationalist. I. A ...

**2**

votes

**1**answer

107 views

### Is torsion submodule of a $p$-adically complete and separated $\mathbb{Z}_{p}$-module closed?

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the ...

**11**

votes

**2**answers

740 views

### Gauss proof of fundamental theorem of algebra

My question concerns the argument given by Gauss in his "geometric proof" of the fundamental theorem of Algebra. At one point he says (I am reformulating) :
A branch (a component) of any algebraic ...

**11**

votes

**2**answers

311 views

### Examples of varieties with many automorphisms acting trivially on co-homology

Let $X$ be a smooth projective variety over the complex numbers. Denote by $Aut(X)$ its automorphism group, by $Aut(X)_0$ the connected component of the identity, and by $G$ the quotient ...

**7**

votes

**0**answers

198 views

### Does Stepanov's method extend to complete intersections?

Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis ...

**4**

votes

**0**answers

193 views

### Reference for Hensel's Lemma in Algebraic Geometry

The following form of Hensel's Lemma in Algebraic Geometry is well-documented in the literature:
$\textbf{Theorem 1}$: Let $R$ be an Henselian local ring with maximal ideal $\mathfrak{m}$, and let ...

**5**

votes

**1**answer

129 views

### Primitive log-divergent graphs and convergence of Feynman amplitudes

To a connected graph $G$, quantum field theory attaches the integral
$$
I_G=\int_{\sigma} \frac{\Omega_G}{\Psi_G^2}
$$ where $N_G$ is the number of edges of the graph, $\sigma$ is the simplex of ...

**3**

votes

**0**answers

246 views

### Why is solving polynomial systems NP hard?

Solving polynomial systems is known to be a NP hard problem; however it is not completely clear to me where this complexity comes from.
My interest is in the case of systems of multivariate ...

**4**

votes

**0**answers

155 views

### Descent of line bundles to the quotient

If a finite group acts $G$ on a variety $X$, consider the quotient $X/G$. I would like to understand which line bundle on $X$ descends to $X/G$. The action is not free. Can anyone direct me to some ...

**6**

votes

**1**answer

256 views

### Smoothness of the “Archimedean special fiber” in Arakelov geometry

If $X$ is a scheme over, let's say, $\mathbb{Z}_p$, one can consider its special fiber obtained by reduction modulo $p$ ans it certainly makes sense to ask if this special fiber is smooth or not.
...

**2**

votes

**1**answer

82 views

### Are the Prym varieties geometrcally nondegenerate subvarieties of the Jacobians?

A subvariety $V$ of an abelian variety $X$ is geometrically nondegenerate if it meets any subvariety of $X$ of dimension bigger than or equal $codim(V)$.
My question is about the Prym varieties as ...

**7**

votes

**1**answer

168 views

### How fine an invariant of a representation is its quotient singularity?

This is a refinement of a question asked on MSE.
Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...

**2**

votes

**1**answer

157 views

### Families of smooth projective varieties over dvr

Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by ...