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

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4
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2answers
226 views

Adjoint semi-simple algebraic groups over non-algebraically closed fields

Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$. On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes algébriques ...
1
vote
0answers
109 views

Relating a fiber of a sheaf and its cohomology

Reading the proof of lemma 4.4.4 in Huybrechts and Lehn Geometry of Moduli Spaces of sheaves I come across an isomorphism relating a fibre of an invertible sheaf and its cohomology, and I really don't ...
4
votes
1answer
153 views

Triple covers of $\mathbb{P}^2$ with fixed branch locus

Let us consider a smooth (complex) cubic surface $X \subset \mathbb{P}^3$ and a general point $p \notin X$. Then it is classically well-known that linear the projection $$\pi_p \colon X ...
5
votes
0answers
152 views

Where can I find Andre's “Cinq exposés sur la désingularisation”?

Many expositions of Popescu's desingularization theorem indicate that an other proof of this theorem can be found in "Cinq exposés sur la désingularisation" by M. Andre, Ecole Polytechnique ...
3
votes
0answers
114 views

Reference for the Hodge polynomial or the Hodge Characteristic

What is the first work that studies, refers to, or mentions the Hodge characteristic? The Hodge polynomial is the unique ring homomorphism $$ P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to ...
3
votes
2answers
212 views

Chern classes and singular hermitian metrics on vector bundles

Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of ...
4
votes
0answers
117 views

On non-vanishing of Milnor K-groups for infinite fields

It is well-known that for $n \geq 2$ and a finite field $k$, the Milnor $K$-group $K_n ^M (k)$ vanishes. I don't know who proved this first, but if curious, you may look at somewhere in Srinivas's ...
0
votes
0answers
112 views

local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...
9
votes
2answers
293 views

Neron models and ramification

I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me: let $E$ be ...
3
votes
0answers
238 views

Restriction of a global moduli functor that admits a coarse moduli space

Let $F:(Sch/k)^{o}\to Sets$ be a functor, where $Sch/k$ is the category of schemes over a field $k$. Suppose that $F$ admits a coarse moduli space, let it be $M$. Consider a $k$-point $x\in M$ (which ...
0
votes
1answer
94 views

Equivalent determinantal hypersurfaces

I have two matrices $A$ and $B$ (of the same order) whose entries are homogeneous polynomial of the same degree. I have that $\det A=0$ and $\det B=0$ define the same hypersurface of $\mathbb{P}^n$ ...
2
votes
0answers
109 views

Computing trace of Frobenius on a local system

Let $E$ be a rank $2$ local system on a curve $X$ over a finite field. Suppose that the Frobenius at $v \in |X|$ has trace $t_v$ and determinant $u_v$. Let $E^{\boxtimes n}$ denote the exterior ...
3
votes
1answer
182 views

Connected-étale sequence for ordinary CM elliptic curves

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$. Suppose that $p$ is ...
4
votes
3answers
122 views

Birationally transforming a quartic elliptic curve

Consider the elliptic curve $$y^2=ax^4+cx^2+dx+f$$ I am aware that there are algorithmic methods for birationally transforming a nondegenerate cubic curve into the Weierstrass canonical form ...
7
votes
2answers
352 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. It is true that if ...
4
votes
1answer
172 views

GCD in polynomial vs. formal power series rings

I'm having problems finding an appropriate reference for this question. Given two elements $f, g \in \mathbb{C}[x_1, \dots, x_n]$, consider their greatest common divisor, $\gcd_{\mathbb{C}[x_1, ...
2
votes
2answers
159 views

Embedding of punctured projective line to abelian variety

Throughout the proof of injectivity of the section conjecture, for example in Appendix B of https://arxiv.org/abs/0809.0017, one uses Mordell--Weil Theorem and for that embeds hyperbolic curve into an ...
2
votes
1answer
215 views

Relative tangent bundle and trivilization, tautological foliation

Let $T_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT_{X}.$ ...
4
votes
0answers
154 views

A crystalline version of an isomorphism of Beauville and Donagi

Let $k$ be an algebraically closed field of characteristic $p>0$ and write $W:=W(k)$ for its ring of Witt vectors. Consider a smooth cubic fourfold $X_{0}\subset\mathbb{P}^{5}_{k}$ and let ...
2
votes
2answers
243 views

Epimorphisms between affine group schemes

Does there exist a simple characterization of epimorphisms between affine group schemes over a field ? Are they faithfully flat morphisms ?
4
votes
3answers
653 views

Hypersurface missing just one point

Let $\mathbb F_q$ be a finite field and $n$ an integer. What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F_q[X_1,\dots,X_n]$ such that the set $Z(f)$ of zeros of $f$ in the ...
2
votes
0answers
116 views

Kodaira fibration and moduli space of Riemann surfaces

Here we mean Kodaira fibration $f: X \rightarrow C$ where $f$ is a holomorphic submersion with maximal rank everywhere, but not a complex fiber bundle map. Such a surface has been constructed by ...
5
votes
2answers
361 views

BSD and generalisation of Gross-Zagier formula

The classical Gross-Zagier formula and the modularity theorem leads to a proof of half-BSD (i.e. an inequality and not equality) for elliptic curves of analytic rank 0. The Gross-Zagier formula gives ...
1
vote
0answers
52 views

truncated arc spaces and intersection numbers

Let $X$ be a variety over a field $k$, by a truncated arc space of order $n$ at $x \in X$ one understands the vector space of morphisms $\mathcal{O}_{X,x} \to k[\epsilon]/(\epsilon^{n+1})$; I will ...
2
votes
0answers
50 views

Preimage of a variety in the incidence correspondence associated to its 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 $X$ ...
5
votes
0answers
147 views

When a proper smooth fibration is isotrivial?

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two ...
4
votes
1answer
164 views

(Etale) fundamental group of quotient singularity $\mathbb{C}^n/G$

I don't know much about (algebraic/etale) fundamental groups, so sorry if this question sounds stupid. I am interested in quotient singularities (quotients $X$ of $\mathbb{C}^n$ by a finite subgroup ...
4
votes
1answer
235 views

Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action

I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far: Let $X=Spec A$ be an affine scheme (after this case is setteled I imagine it ...
7
votes
1answer
204 views

Archimedean fibers “intersecting” curves on arithmetic surfaces

Let's fix a number field $K$ with its ring of integers $O_K$. Moreover consider an arithmetic surface $f:S\to \text{Spec } O_K$. For every archimedean place $\sigma$ in $K$, $K_\sigma$ is the ...
6
votes
0answers
319 views

Competing notions of étaleness

I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate. Here is a list of ...
10
votes
0answers
696 views

Is it worth the efforts to read books/papers written in Weil's algebraic geometry language

There is much important work written in Weil's language of algebraic geometry rather than schemes (besides Weil himself, I can think of Shimura, Neron immediately). My question is: is it worth the ...
4
votes
1answer
300 views

Are stably birational varieties birational?

We say that two (irreducible) algebraic varieties $X$ and $Y$ are stably birational if $X \times \mathbb{P}^n$ is birational to $Y\times \mathbb{P}^n$ for some $n\ge 0$. The natural question is then ...
1
vote
0answers
101 views

On the maximal powers of $q$ which arise in a quantum product

Let $X=G/P$ be a generalized flag variety (where $G$ denotes a connected, simply connected, semisimple complex linear algebraic group and $P$ a parabolic subgroup). In this paper by Fulton and ...
13
votes
1answer
464 views

$p$-adic Bott periodicity?

The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...
3
votes
0answers
167 views

Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions). The ...
3
votes
0answers
77 views

Effective divisor in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-classes

Does anybody knows an effective class in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-coefficients? The standard references; Logan, Farkas or Brill-Noether divisors have all non-negative ...
1
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0answers
59 views

A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...
4
votes
1answer
111 views

How to write the map $ℂ[G/U]↪ℂ[B]$ explicitly?

Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat ...
8
votes
1answer
418 views

Cohomology theory “from” Grothendieck's six operations?

How, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos? I would like some ...
2
votes
0answers
95 views

Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...
6
votes
0answers
164 views

Semi-continuity of intersection numbers

I always trusted the following quite vague statement: If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say ...
2
votes
1answer
225 views

Field of definition of an algebraic set

I find this definition in Silverman's book, The Arithmetic of Elliptic Curves: an algebraic set(in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in ...
1
vote
0answers
118 views

globally well-defined holomorphic vector field on a curve $y^N = x^2 - z^2$

Let us start with a multiple cover $C$ of the $x$-plane branched at $z$ and $-z$, and so described by an equation $y^N = x^2 - z^2$. For $N=2$, it is known that there are globally-defined ...
3
votes
0answers
74 views

Bott-type vanishing results for the weighted Grassmannian wGr(2,5)

If $G=Gr(k,n)$ denotes the Grassmannian of k-dimensional subspaces in $V= \mathbb C^n$, representation theory gives us a Bott-type result for the cohomology groups $H^q(G, \Omega^p(k))$ of the twisted ...
1
vote
2answers
326 views

Perverse sheaves and tensor product

If $X$ is a connected algebraic variety of finite type over $k$ (with $k$ a field of positive characteristic) of dimension $d$, and if $\mathcal{F}$ and $\mathcal{G}$ are perverse sheaves on $X$ so ...
7
votes
4answers
442 views

Picard groups of quartic K3 surfaces

Does anyone know where I can find examples of quartic K3 surfaces for which the Picard group is known? I'm really interested in examples where there are explicit constructions of the divisors ...
1
vote
0answers
87 views

Does the link of a hypersurface singularity determine its analytic type?

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a ...
1
vote
0answers
69 views

connectedness of semi algebraic sets

We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, ...
2
votes
1answer
146 views

Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?

Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$. Consider the condition $$\omega_C\simeq N_{C/S}$$ For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and ...
5
votes
2answers
592 views

Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves" I hope I'm not distorting his phrase, can someone ...