3
votes
0answers
118 views

P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...
0
votes
1answer
166 views

Are period domains ever contractible

Which simply-connected period domains are contractible? Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety? Are these contractible?
4
votes
2answers
179 views

Branch locus of a 6:1 cover of the grassmannian G(1,3)

Given a general quartic surface $S$ in $\mathbf{P}^3$, there is a natural 6:1 surjective map $\phi: Hilb^2(S) \to G(1,3)$ sending $\{P,Q\}$ to the line through them in $\mathbf{P}^3$. Can you ...
0
votes
0answers
129 views

Foliation over characteristic positive

Ekedahl wrote about foliation in characteristic positive, over the field $/frac{Z}{pZ}$ as a subsheaft of the tangent sheaft, that is closed over involution and $p$-power, My question is if there ...
5
votes
1answer
234 views

stackification commutes with finite limits?

Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes. Let us denote the category of stacks ...
0
votes
1answer
278 views

Geometrically connected curve

What is the definition of a geometrically connected curve?
3
votes
1answer
240 views

Hilbert scheme of 2 points on an elliptic curve

The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$. What can we say about the ...
5
votes
1answer
279 views

kapranov's realization of $\overline{M}_{0,n}$ over other fields

Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, ...
6
votes
1answer
304 views

question about higher geometric stacks

I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O_X) such that Hom(X, Spec(A))= ...
4
votes
1answer
278 views

Invariant lattice of algebraic surface.

Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or ...
4
votes
0answers
314 views

A question on an intuitive way to look at stacks

I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...
0
votes
2answers
458 views

What's the meaning of pencils in birational geometry?

I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi_{|D|}(X)\geq 2$ and even ...
2
votes
1answer
292 views

T^i functors are isomorphic for analytically isomorphic isolated singular points

I've been having trouble proving the following: Let $B$ and $B'$ be local rings, essentially of finite type over $k$, having isolated singularities at the closed points. Suppose that they are ...
2
votes
1answer
275 views

Frobenius base change of etale maps

Let $A$ be a characteristic $p>0$ commutative ring. Let $B$ be a finitely presented etale $A$ algebra i.e. $$ B=A[x_1,\ldots,x_n]/(f_1,\ldots,f_n), $$ with $det(\frac{\partial f_i}{\partial ...
1
vote
0answers
360 views

elementary exact sequence of normal sheaves

Let $Z \subset Y \subset \mathbb{A^n}$ be a smooth subvarieties of $\mathbb{A^n}$. I'm trying to show that there is an exact sequence of normal bundles. $0 \rightarrow N_{Z/Y} \rightarrow N_{Z} ...
5
votes
0answers
178 views

Does every stack with a connection admit an atlas with a connection?

Dear all, Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...
2
votes
1answer
412 views

A real algebraic curve in $\mathbb{R}^3$ is the intersection of zero sets of polynomials in $\mathbb{R}[x,y,z]$. Can we choose the polynomials in a way, that, seen in $\mathbb{C}[x,y,z]$, the intersection of their zero sets is a complex alg. curve?

By "algebraic curve" here (both in $\mathbb{R}^3$ and in $\mathbb{C}^3$), I do not only mean that the dimension of the set as an algebraic variety is 1, but also that its dimension is 1 in a ...
3
votes
1answer
160 views

bounding the probability that a polynomial is near 0

Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
2
votes
2answers
372 views

branch points of modular parametrization of an elliptic curve

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to ...
1
vote
2answers
264 views

Abelian subgroups of ball quotient

Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can ...
4
votes
1answer
425 views

Localizability of differential operators a la Grothendieck

Hello, Maybe this question is trivial, so sorry Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1). Then we can define the module of differential operators $D^{\leq n} ...
0
votes
0answers
202 views

Properties of morphisms induced by divisors on curves

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim ...
1
vote
1answer
349 views

two polynomials

If $p,q:\mathbb R^3\rightarrow\mathbb R$ are two polynomials, such that $\{p=0\}\cap\{q=0\}$ is two-dimensional, does it follow that $p$ and $q$ have a common factor? (I believe it does.) How to prove ...
1
vote
0answers
136 views

Morphisms, inducing isomorphisms on global functions

Let $f:X\to Y$ be a surjective morphism of complex varieties with $Y$ affine. Assume that every fiber of this morphism has the property that all the global functions are constant. What else do I need ...
3
votes
1answer
516 views

moduli of vector bundles on a surface

Let $S$ be a smooth projective surface with an ample divisor $X\subset S$. Consider the moduli stack of vector bundles $F$ on $S$ such that 1) $c_1(F)=0$ 2) $c_2(F)=n$ 3) The restriction of $F$ to ...
3
votes
1answer
370 views

Connecting points on a variety by the image of a nonsingular curve

In Hartshorne's proof of a result of Igusa (see III, 9.13 of Hartshorne) he claims without proof that any two closed points on a variety can be connected by the image of a nonsingular curve, or by a ...
9
votes
2answers
1k views

Roadmap to Computer Algebra Systems Usage for Algebraic Geometry

I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more. ...
7
votes
3answers
433 views

Symmetric sequence of blow-ups for the Fulton-MacPherson compactification

Let $X$ be an algebraic variety and $X[n]$ be the Fulton-MacPherson compactification of the configuration space $F(X,n)$ introduced in the paper "A compactification of configuration spaces". In this ...
0
votes
1answer
206 views

Chow group of a fiber product of grassmann bundles

Let $X$ be a smooth projective variety and $E\longrightarrow X$ a vector bundle of rank $n$. For any $0\leq k\leq n$ the associated Grassmann bundle $G_k(E)\longrightarrow X$ yields and we have the ...
2
votes
0answers
211 views

volume under induced metric of preimage set of a regular value under a polynomial map

I am interested in the following kind of polynomial map: $$ f: \mathbb{T}^k \to \mathbb{R}^n$$ where $f$ is a polynomial map of a certain maximum degree $d$, in the sense that if we imbed ...
2
votes
1answer
220 views

Image of the Abel map for a hyperelliptic curve

Let $C$ be a smooth projective curve over some field (algebraically closed if necessary), ${\rm Pic}^{d}(C)$ the Picard variety of degree $d$ divisor classes on $C$, $C^{(d)}$ the $d$th symmetric ...
3
votes
2answers
382 views

Vanishing of Self-Ext groups of vector bundles

Let $E$ be a rank two vector bundle on $\mathbb{P}^n$. Assume that $\text{Ext}^1(E, E)=0$. Will $\text{Ext}^2(E, E)$ be zero? Why? Any geometric explanation (in terms of deformation theory?)? Edit: ...
2
votes
3answers
261 views

Implement intersection products

I am doing a counting problem, and it comes to compute intersection products ( Chow ring ) on some varieties. Is there any computer algebra that deals with this?
3
votes
3answers
379 views

Nature of Invertible Sheaves in which there are no global sections.

EDIT: Let me try to make the question clearer. Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
4
votes
2answers
582 views

Non-finitely generated ring of regular functions

It is remarked in Shafarevich's Basic Algebraic Geometry 1 that Rees and Nagata constructed examples of quasiprojective varieties such that the ring of regular functions is not finitely generated, but ...
20
votes
6answers
1k views

Formal consequences of Riemann-Roch (multiple answers welcome)

This question aims to pin down what Riemann-Roch can tell us about a divisor on a curve, without any "geometric thinking". It can be annoying to wonder if there is some clever trick you're missing ...