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

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4
votes
1answer
468 views

Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...
2
votes
0answers
118 views

Varieties with Chow groups supported in positive codimension: examples and properties?

In their 1983 paper "Remarks on correspondences and algebraic cycles" Bloch and Srinivas proved several interesting properties of smooth proper varieties (over universal domains) whose Chow groups of ...
1
vote
0answers
93 views

G-invariant vector bundle over schemes?

Take X to be a scheme over a (say algebraic closed) field k and G a group scheme act on X. I want to get more convenient knowledge about G-invariant vector bundles over X. I'm inspired by this ...
4
votes
1answer
166 views

$n$ columns of a specific “infinite” Vandermonde matrix always linearly independent? [closed]

Consider the "infinite" Vandermonde matrix $$ V (x_1, x_2, \ldots , x_n) = \begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\ ...
1
vote
0answers
104 views

Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and $$ X_n=\underbrace{ X\times_S ...
2
votes
1answer
206 views

On a property of the Grothendieck group of a smooth projective curve

Let $K$ be a complete DVR of characteristic $0$, $X$ a smooth projective curve over $K$. Denote by $K^0(X)$ the Grothendieck group of locally free sheaves on $X$ and by $\mbox{det}$ the natural group ...
0
votes
0answers
109 views

Descent datum for a line bundle

Let $\pi:C \to \mathbb P^1$ be a double cover branched at $r$ points. To understand the theory of descent better, I would like, if possible, to construct by hands the descent datum of a line bundle ...
15
votes
3answers
1k views

Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...
2
votes
0answers
140 views

l-adic cohomology and perverse sheaves

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...
1
vote
0answers
85 views

Intuition for the structure theorem for connected solvable algebraic groups over an algebraically closed field of positive characteristic

In the theory of algebraic groups one of the fundamental results is the structure theorem for connected solvable groups. I never understood the proof in any of the standard textbooks, so I just moved ...
3
votes
1answer
176 views

Spec of an injective ring map contains minimal primes in its image?

Let $f\colon A \rightarrow B$ be an injective ring homomorphism. One knows (from EGA I, 1.2.7 or elsewhere) that the image of $\mathrm{Spec}(f)$ is dense. Does that image necessarily contain all the ...
2
votes
0answers
80 views

question on n-th root coverings

Let $X$ be a smooth projective complex variety and $D \subset X$ an effective divisor. Assume that there is a line bundle $L$ such that $L^n=\mathcal{O}(D)$ for some integer $d \geq 2$. Suppose $s ...
0
votes
0answers
182 views

“Spreading out” locally free sheaves

Let $Y$ be a regular surface flat, projective over $R$, where $R$ is complete DVR. Let $X$ be the generic fiber and $F$ be a locally free sheaf on $X$. We know that if the rank of $F$ is $1$ then we ...
0
votes
0answers
61 views

birational equivalence of linear sections of algebraic varieties

Let $X$ be an irreducible algebraic variety and suppose that $L$ is a linear space defined by the linear forms $l_1,l_2,\ldots,l_k$. I want to study $L\cap X$. I would like to know whether the ...
3
votes
0answers
136 views

Cancellation and splitting theorems for vector bundles etc over schemes

It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...
0
votes
0answers
109 views

A question on a toric singularity

Recently I was told that the threefold singularity $$ x^2+y^2+z^2+w^{2n}=0 $$ in $\mathbb{C}^4$ is a toric variety. How can I see this? What are the generators of the toric fan? Moreover, is it true ...
1
vote
1answer
151 views

pencil of quadrics consisting of singular quadrics

A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if: (1) quadrics in $l$ have a common singular point; or (2) quadrics in $l$ contain a common ...
0
votes
0answers
174 views

Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...
10
votes
1answer
588 views

Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?

I am wondering if there is a multi-dimensional analog of the Birch/Swinnerton-Dyer (BSD) conjecture. The recent famous result inching toward resolution of that conjecture is: Bhargava, Manjul, and ...
1
vote
0answers
74 views

Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers. Can one classify all reductive group schemes over $C$? Certainly, you have the trivial ones (coming from pulling-back ...
4
votes
1answer
112 views

Canonical bundle of moduli space of rational curves and automorphisms

Let $\overline{M}_{0,n}$ be the usual Deligne-Mumford compactification of $M_{0,n}$ the moduli space of smooth $n$-pointed rational curves. The canonical divisor $K_{\overline{M}_{0,n}}$ can be ...
2
votes
0answers
62 views

Criterion for the existence of finite locally free resolution

Let $X$ be a projective variety over an algebraically closed field $k$, $S$ be a $k$-scheme, $E$ be a coherent sheaf on $X \times_k S$, flat over $S$. We know that if $X$ is smooth then $E$ has a ...
1
vote
0answers
132 views

Is there a D-module theory in characteristic p>0

everyone. In characteristic 0 we have a good theory of D-modules. In particular, we have a formalism of Grothendieck's six operators in the derived category of holonomic D-modules and Riemann-Hilbert ...
2
votes
0answers
103 views

subschemes of abelian scheme over artinian basis

Let $R$ be an artinian thickening of a field $k$. Denote with $S=Spec(R)$. Let $A$ be an abelian scheme over $S$. Let $X$ be a closed, reduced, equidimensional subscheme of the special fiber $A_k$. I ...
2
votes
1answer
121 views

convergence of L-functions of curves

Let $C$ be a smooth projective curve over $\mathbb{Q}$. Its associated L-function is defined by $$ L(C, s)=\prod_{p \text{ prime}} L_p(C, s), $$ where, if $p$ is a prime of good reduction, ...
7
votes
0answers
130 views

Vector bundles with flat connections

Let $V$ and $W$ be varieties defined over a number field, and let $f: V \rightarrow W$ be a map. Suppose that the induced map on the complex points of $V$ and $W$ induces an isomorphism on fundamental ...
2
votes
0answers
102 views

Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
4
votes
0answers
71 views

Looking for some abelian surface fibration

Do you know of any explicit smooth complex projective threefold $X$ with an Abelian surface fibration over $\mathbb{P}^1$ such that $K_X = [-F]$ where $F$ is the fiber class divisor. I am not ...
3
votes
1answer
148 views

Subspaces of Grassmannian under Plucker embedding

The Grassmannian $G(k,V)$ is the set of all $k$-dimensional subspaces of a vector space of dimension $n$. It can be embedded inside $\mathbb{P}(\bigwedge^k V)\cong \mathbb{P}^{\binom{n}{k}-1}$ as a ...
4
votes
0answers
86 views

Strategy to prove formula for top chern class from knowlege of chern character

I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up. I have a sequence of (smooth, complex, rationally connected) ...
8
votes
2answers
322 views

“Forms” of quadrics

The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over ...
6
votes
1answer
202 views

Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?

My question is: Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by ...
2
votes
1answer
140 views

The stable general linear group in algebraic geometry

Let $A$ be a $ \mathbb{C}$-algebra. Then we have the following increasing union: $$ {\rm GL}_1(A) \subseteq {\rm GL}_2(A) \subseteq {\rm GL}_3(A) \subseteq \dots \subset {\rm GL}(A) $$ We call $ {\rm ...
4
votes
2answers
213 views

Non trivial family of hyperelliptic curves

Let $X$ ba a smooth hyperelliptic curve of genus $g$, and let $f:X\rightarrow X$ be the hyperelliptic involution. Consider a $K3$ surface $S$ with an involution $g$ without fixed points. The quotient ...
3
votes
0answers
78 views

birational classification of rationally connected 3-folds

What is the birational classification of (smooth projective) rationally connected 3-folds (over algebraically closed fields of characteristic $0$ or even $\mathbf{C}$, if $\mathrm{char}(k) = p > ...
2
votes
1answer
153 views

discriminant of smooth quartic del Pezzo surface in $\mathbb{P}^4$

I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves. Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics ...
1
vote
0answers
201 views

Two questions on canonical line bundle over $\mathbb{C}P^{n}$

The canonical line bundle over $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic ...
2
votes
1answer
239 views

Stalks of étale sheaves

I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...
1
vote
1answer
134 views

L-function of twist

I'd like to ask the following easy question, since I can't find a reference. Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...
10
votes
1answer
561 views

Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$: $$ \begin{cases} 2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\ ...
1
vote
0answers
212 views

Learning roadmap in Algebra [closed]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
2
votes
1answer
82 views

u-Invariants of p-adic function fields

In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have ...
3
votes
0answers
107 views

Singularities in mixed characteristic

Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the ...
3
votes
1answer
166 views

Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...
2
votes
0answers
60 views

Algebraic approach to showing trigonometric equations have no solution

I have very little background in algebra and algebraic geometry, so please bear with me. I am trying to show that certain systems of trigonometric polynomial equations generally have no solution. One ...
5
votes
1answer
129 views

p-adic L-function of curves

Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators. Is there a p-adic ...
3
votes
2answers
194 views

Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...
1
vote
1answer
99 views

Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...
4
votes
1answer
185 views

transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number $$ \beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...
1
vote
0answers
73 views

plane curves with two points of high multiplicity

Let $\mathcal{C}$ be an irreducible plane curve in $\mathbb{P}^2_\mathbb{C}$ of degree $d$. Let $D$ be a quartic with three irreducible components with normal crossing singularities, i.e. a conic and ...