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

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votes

**1**answer

82 views

### Linear systems and 2-torsion shifts on hyperelliptic curves

Let $C$ be a hyperelliptic curve of genus $g$ and let $D$ be a divisor on $C$ of degree $g+1$. Assume that the linear system $|D|$ is base-point-free. Now add a $2$-torsion point $[E]$ to $D$. I would ...

**1**

vote

**0**answers

57 views

### On convex hull of algebraic curve

For any given tuple of quadratic functions $(f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))$, the following set forms a algebraic curves
$$
S_{f_1,f_2,f_3}=\{(f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))|x^2+y^2+z^2=1,x,y,z\...

**2**

votes

**1**answer

161 views

### Regarding a conjecture Fogarty proposed

In a paper by Fogarty titled "Algebraic Family On An Algebraic Surface,"
he conjectured that $\bf Hilb^n(\mathbb P^N)$ is always variety-- reduced and irreducible.
Is this still a conjecture; any ...

**4**

votes

**1**answer

247 views

### Algebraicity and non-algebraicity of leaves of the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of
$D$, the restriction of $\omega$ to $D$ has one-...

**4**

votes

**2**answers

235 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 ...

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vote

**0**answers

124 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

**1**answer

156 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 \...

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**0**answers

158 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 ...

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votes

**0**answers

121 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 \mathbb{Z}[u,v,u^{...

**3**

votes

**2**answers

241 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

**0**answers

119 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 ...

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votes

**0**answers

128 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

**2**answers

307 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

**0**answers

255 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

**1**answer

100 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 $\mathbb{P}^n$ (...

**3**

votes

**1**answer

186 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 ...

**5**

votes

**3**answers

157 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

**2**answers

508 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

**1**answer

181 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, \dots,...

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votes

**2**answers

166 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

**1**answer

238 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 $PT_{X}.$
...

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votes

**0**answers

161 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 $F(X_{0})...

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votes

**2**answers

256 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

**3**answers

692 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 ...

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votes

**0**answers

123 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

**2**answers

382 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 ...

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vote

**0**answers

54 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 ...

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votes

**0**answers

53 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

**0**answers

151 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 ...

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votes

**1**answer

172 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

**1**answer

252 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 ...

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votes

**1**answer

213 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 ...

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votes

**0**answers

353 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

**0**answers

706 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

**1**answer

315 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

**0**answers

110 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

**1**answer

500 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

**0**answers

173 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

**0**answers

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 ...

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**0**answers

60 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 ...

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votes

**1**answer

113 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 ...

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votes

**1**answer

438 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 ...

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votes

**0**answers

97 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 ...

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**0**answers

171 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 ...

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votes

**1**answer

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 $K[X]=K[...

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**0**answers

119 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 ...

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votes

**0**answers

75 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 ...

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vote

**3**answers

500 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

**4**answers

454 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 ...

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vote

**0**answers

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 ...