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

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2
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0answers
97 views

Galois descent for a non-Galois extension

Suppose that $k$ is an algebraically closed field of characteristic $p > 0$ and $E/k$ is a supersingular elliptic curve equipped with a full level $N$ structure $\phi$ for some $N \ge 3$ that is ...
8
votes
1answer
310 views

Automorphisms of generic complete intersections

This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions. To state the question, let $r \geq ...
14
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1answer
2k views

Mazur secret Bourbaki report “Analyse p-adique”

Does anyone happen to know if a scan of Mazur's report exists, and, if so, where to find it? It appears in the references for Katz's "Higher congruences" and "Eisenstein measure" papers.
6
votes
1answer
201 views

An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal

We know that for a curve $X$, any object $\mathcal{E}^{\bullet}$ in the derived category $D^b_{\text{coh}}(X)$ is formal, i.e. $\mathcal{E}^{\bullet}$ is quasi-isomporphic to the direct sum of its ...
1
vote
1answer
182 views

Volume of a GIT quotient of projective lines

Say $X= \mathbb{P^1}\times \cdots \times \mathbb{P}^1$ is a product of $n\geq3$ lines. Let the group $G=\text{SL}(2)$ act on $X$ diagonally, and let $\mathcal{L} = \mathcal{L}(a_1,\ldots,a_n)$ be the ...
2
votes
1answer
314 views

A question about an intersection number

Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
-2
votes
0answers
67 views

what are the maps in this exact sequence? [migrated]

Let $X$ be a smooth projective variety over the complex number and $D$ a divisor with simple normal crossings. Denote by $U$ the complement of $D$ in $X$. Then one has a long exact sequence in ...
1
vote
0answers
63 views

degenerate abelian surfaces

I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...
1
vote
0answers
51 views

Normalization (integral closure) of $\mathbb Z_p[x]$ in function field of a curve to obtain Model of curve

I want to follow this construction of a normal model of a curve: Let $p\neq 2,3$ and $Y\to \mathbb P¹$ be a smooth projective curve over $\mathbb Q_p$ with function field $L/\mathbb Q_p(x)$ e.g. ...
-2
votes
0answers
84 views

pullback of global sections with respect to an automorphism of schemes [migrated]

Let $X$ be a projective scheme and $\sigma:X\to X$ an automorphism of $X$. Is there a natural pullback of global sections map $H^0(X,\mathcal{F}) \to H^0(X,\sigma^*\mathcal{F})$ for ...
0
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0answers
63 views

How to understand/analyze vanishing cycles and fibers of 6 dimensional Lefschetz fibration?

Say you have a polynomial $f(x,y,z,m)=x^3+y^3+z^3+m^3$ where $x,y,z,m \in \mathbb{C}$. Consider the Lefschetz fibration from $\{f=\mu\} \cap \{ |m|\leq \delta \}$ to $m$ for suitably small $\mu$ and ...
0
votes
0answers
97 views

How can every divisor be reached by a sequence of blow-ups?

The following is a result of Zariski [cf. Lemma 2.45 of Birational Geometry of Algebraic Varieties]. $X$ : an algebraic variety over a field $k$. $(R,m)$ : a DVR of the quotient field $K(X)$ ...
12
votes
0answers
207 views

The topos for forcing in computability theory

My understanding is that forcing (such as Cohen forcing) can be described via a topos. For example this nlab article on forcing describes forcing as a "the topos of sheaves on a suitable site." My ...
1
vote
0answers
127 views

Coherence of $\mathcal O_X[T]$

Let $X$ a complex manifold, and $\mathcal O_X$ the sheaf of holomorphic functions. Oka Coherence Theorem states that $\mathcal O_X$ is coherent (as $\mathcal O_X$-module). How to prove that also the ...
0
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0answers
169 views

When an Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical. My question is When an Spherical variety is $K$-stable? Is ...
1
vote
0answers
70 views

Hyperspecial parahoric group schemes/Chevalley groups

Let $G$ be a simple group over $k=\mathbb{C}$, $A=k[[t]]$, $K=k((t))$, and consider the group $G(K)$. This group is a split reductive group over a local field, and therefore the results of Bruhat and ...
1
vote
0answers
128 views

map in K-theory

I have a very stupid question on a map I have seen in K-theory. The situation is as follows: $X$ is a smooth variety over a field $k$ and $\iota: Z \to X$ is the inclusion of a smooth closed ...
1
vote
0answers
46 views

An explicit description of the Torelli spaces of pointed genus 2 Riemann surfaces

In [N], there is a nice and very explicit description of the Torelli space ${\rm Tor}_{1,n}$ of $n$-pointed elliptic curves, for any $n\geq 1$: $$ {\rm Tor}_{1,n}=\left\{ \big(\tau, ...
0
votes
1answer
280 views

Stable Vector bundles

Let $C$ be smooth curve, and let $F$ be a stable rank 2 vector bundle of degree equal to $2c+1$, $c\in\mathbb{N}$, and fix a point $p\in C$: Can one choose an epi-morphism $u:F\rightarrow \mathbb ...
3
votes
0answers
84 views

Torsors under the group scheme over projective line

Consider the group scheme $\mathcal T$ over $\mathbf P^1$ given locally (variable $t$) by the equation $x^2 - f(t)y^2 = 1$ where $f(t)$ is a polynomial of degree $r$ with distinct roots (assume that ...
11
votes
1answer
341 views

Schemes over topological rings

I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ...
-2
votes
0answers
25 views

What does it mean to tensor with Q [migrated]

At our algebraic geometry seminar I often hear that something is 'tensored with Q', eg a ring of endomorphisms. This phrase seems to have some intuitive meaning that I don't know. What does 'tensoring ...
1
vote
0answers
113 views

Hermitian metric on conic Kaehler-Einstein setting

I have a technical question : Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler ...
5
votes
0answers
114 views

LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...
2
votes
2answers
231 views

Heuristics for 2-morphisms of (algebraic) stacks

For topological spaces and simplicial sets one can consider each pair of parallel morphisms $f,g:X\rightrightarrows Y$ as equipped with a set of 2-morphisms given by homotopies $H:f\simeq g$ (let's ...
3
votes
0answers
92 views

ramified principal prime ideals in Artin-Schreier extension

Let $q$ be a power of $2$ and $K$ be the quadratic extension of $\mathbb F_q(T)$ defined by $K:=\mathbb F_q(T)[y]$ where $y^2+y=f(T)\in\mathbb F_q(T)$. Put $\mathcal O_K$ the integral closure of ...
1
vote
1answer
211 views

How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?

For Calabi-Yau variety $X$ which is a complete intersection $$ f_1=f_2=\ldots=f_r=0 $$ in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the ...
1
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0answers
47 views

cone and its scaled image

Let $C$ be a polyhedral cone in $\mathbb{R}^m$ defined by $C = \{R y : y \in \mathbb{R}^m_+\}$ and $R\in\mathbb{R}^{m\times m}$. Let $S: \mathbb{R}^m \to \mathbb{R}^m$, be a scaling map, i.e. $S = ...
5
votes
2answers
845 views

Physicist trying to understand modern mathematics

I'm a physicist trying to gain a deep understanding of mathematics that is required for my work.I intend to specialize in string theory which is a very math intensive branch of theoretical physics ...
3
votes
0answers
143 views

When can a blow-down be pushed out?

I am interested in constructing a morphism out of a blown down variety. Let $V$ be a scheme, $U\hookrightarrow V$ an open immersion. Let $\widetilde V$ be a blow-up of $V$, $\widetilde U$ its ...
2
votes
0answers
84 views

Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points

Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
1
vote
1answer
78 views

About 3-fold log canonical singularity

As far as I know, log canonical surface singularities were classified. How about higher dimensional case? I especially want to know whether given 3-fold singularity is log canonical or not. Let $f$ ...
4
votes
1answer
169 views

Vanishing theorem for big divisors

Let $X$ be a projective, smooth variety over $\mathbb{C}$, and let $D$ be a irreducible, big Cartier divisor(notice, I do not assume nefness). Then is it true that $${\rm{H}}^1(X, K_X + D) = 0\quad?$$ ...
3
votes
2answers
278 views

A reference for “an algebraic space is a scheme iff its reduction is”?

It seems to be a known fact that an algebraic space is a scheme if and only if its associated reduced closed subspace is a scheme. For instance, this is used in Chai-Faltings in proving that the dual ...
7
votes
1answer
269 views

Is there a description of the moduli space of elliptic surfaces?

In this question elliptic surface means a smooth projective complex surface $X$, such that there is an elliptic fibration $\pi \colon X \to C$. (I.e., there is a curve $C$ and a proper map $\pi$, such ...
3
votes
0answers
125 views

Linearized action of a torus and orbits in Alexeev's construction

I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I'm having quite a hard time.. So this time my question is the following: Let's consider a stable toric pair ...
17
votes
1answer
571 views

On Grothendieck's idea on his Standard Conjecture B

Let me recall the Standard Conjecture B (see [1,2] below): The $\Lambda$-operation of Hodge theory is algebraic. It more or less says that the partial inverse to “cupping with the class of a ...
3
votes
0answers
164 views

Bruhat decomposition of $G/Q$

Let $G$ be a semisimple algebraic group over $\mathbb C$, $T$ be a maximal torus and $B$ be a Borel subgroup of $G$ containing $T$. Let $R^+$ be the set of positive roots with respect to $B$. Let $Q$ ...
5
votes
0answers
155 views

homology theory for affine and projective algebraic sets?

Given $f_1,\ldots,f_r\in K[x_1,\ldots,x_m]$, resp. homogeneous $f_1,\ldots,f_r\in K[x_0,\ldots,x_m]$, is there a chain complex built from these polynomials, such that any polynomial map $\varphi\!: ...
2
votes
1answer
152 views

Reference request: log Fano varieties

I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano. Thanks a lot.
3
votes
0answers
180 views

Hypersurface with singularities

I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?). It was said ...
-1
votes
1answer
87 views

Can a rational map be extended without using resolution of indeterminacies? [closed]

Suppose I have a finite morphism C --> D, where C is an open subvariety of some projective variety C' and D is open in D', also projective. Thus there is a rational map from C' to D'. Is there a way ...
0
votes
0answers
108 views

Isomorphism in a derived category of chain complexes with rational coefficients

Let $C$ be the category of quasi-projective schemes over a base field $k$ (maybe I will need some assumptions on $k$). Let $Ab(C_{\tau})$ be the category of Abelian sheaves on a site $C_{\tau}$, where ...
0
votes
0answers
129 views

Ring of integers in Artin-Schreier extension

Question put in mathstackexchange but received no answer. It is well-known( see Goldschmidt book: Algebraic Functions and Projective Curves) that for $q$ a power of $2$ a quadratic separable ...
93
votes
1answer
4k views

What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title. Recently, there has been a flurry of new discussion ...
0
votes
0answers
163 views

projective map from $\overline{\mathcal{M}}_{0,n}$

Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). ...
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votes
1answer
160 views

Direct image of structural sheaf [closed]

I am sorry if my question is not of high level!! Let $\pi:X\rightarrow Y$ be a double cover where $X$ and $Y$ are projective smooth curves. Is it true that $R^1\pi_*\mathcal O_X=0$ ? Why ? Thanks ...
0
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0answers
146 views

Twisting sheaf of Serre

I'm sorry if my question is rather trivial, but I can't figure it out.. Given $A$ a ring and $P=Proj(A[X_0,\cdots,X_n])$, I know that $\oplus_n H^0(P,\mathcal{O}(n))=A[X_0,\cdots,X_n]$. This equality ...
1
vote
0answers
163 views

Polynomial existence over finite field

Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$. Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$). Denote ...
5
votes
0answers
175 views

Conjugation of the quotient of $SL(n,\mathbb{C})$ by a finite subgroup

EDITED Let $G={SL}_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$. Let $H\subset G$ be a finite subgroup. Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety. ...