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

learn more… | top users | synonyms (1)

1
vote
0answers
33 views

References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case. I am ...
-1
votes
0answers
37 views

Complementary polynomials

Denote $S=\{0,1\}^n$. $\mathsf{MLP}_{d,n}=\{p\in\Bbb R[x_1,\dots,x_n]:p\mathsf{\mbox{ is mutilinear with total degree}}(p)=d\}$. Is there an $n\geq d^2+1$ such that there exists distinct polynomials ...
0
votes
1answer
79 views

Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...
5
votes
0answers
73 views

Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.) Under which ...
0
votes
0answers
54 views

Moduli space of points - Gorenstein ideal

I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d). I'm more interested in the algebraic point of ...
1
vote
0answers
80 views

Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$. If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
11
votes
1answer
346 views

Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components in which all the rational points lie only on one component? Concrete examples are really appreciated.
0
votes
0answers
56 views

What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex $$ V ...
-2
votes
0answers
74 views

Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris [on hold]

Let π:C′→C an unramified double cover of a complex Riemann surface C of genus g. With the symbol Nmπ we mean the norm application that takes a meromorphic function on C′ named h and produce a ...
-2
votes
0answers
63 views

involution of a riemann surface [on hold]

I've have found in the book GEOMETRY OF ALGEBRAIC CURVES by Arbarello, Cornalba, Griffiths that if π:C′→C it's a double unramified cover of a complex riemann surface named C that we can define the ...
0
votes
0answers
53 views

Circle squarer and solution of polynomial equations [on hold]

Given a circle squarer, i.e. something that gives a length of $\pi$ given an unit length and vice versa, and a straightedge and compass, is it possible to solve any polynomial equation with rational ...
4
votes
0answers
77 views

Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...
2
votes
0answers
52 views

Does the invariant from resolution of singularities provide a Whitney stratification?

The topic of Whitney stratifications came up in a lecture, and the general procedure in the examples was to decompose the singular locus of the variety into the strata starting with the "worst" ones. ...
0
votes
0answers
131 views

What is a Beilinson spectral sequence?

I'm writing to ask just a question. I would like to understand better what is the Beilinson's spectral sequence and how it can be used. Is there any useful and clear reference you advice to someone ...
1
vote
1answer
271 views

Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
3
votes
1answer
138 views

Verlinde Formula and Theta Function Identities

The paper Fusion rules and modular transformations in 2D conformal field theory by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just ...
1
vote
1answer
132 views

rational point of a curve [on hold]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...
2
votes
2answers
185 views

Rigid curves, and the “richness” of their homology class

Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H_2(X,\mathbb Z).$$ I have a very vague question about this situation: Q. ...
3
votes
1answer
168 views

Abelian varieties as analytic manifolds

Assume we have an Abelian varieties over the p-adic numbers, namely $ k=\mathbb{Q}_p$. Then the question is whether $A(k)$, the rational points over $k$, will form a p-adic analytic manifold. I am ...
-2
votes
0answers
63 views

Existence of general element in module case [on hold]

Let $(R,m)$ be a noetherian local ring. Let $a_1,a_2,a_3\in R\oplus R$ and $S=R[X_1,X_2,X_3].$ Then is it true that if $z=a_1X_1+a_2X_2+a_3X_3\in S\oplus S$ then $\frac{S}{zS}\cong S[Y].$
10
votes
1answer
183 views

Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$. Must $V$ have a solvable point? The variety $V$ is assumed ...
6
votes
1answer
283 views

Étale cohomology versus classical cohomology

Let $X$ be an algebraic variety over $\mathbb{C}$. If $X$ is smooth, the étale cohomology $H^p_{\textrm{ét}}(X,\mathbb{Z}/n)$ is isomorphic to the singular cohomology ...
2
votes
1answer
106 views

Flatness of a morphism of complex analytic spaces

Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced. ...
2
votes
1answer
95 views

Intersection of two real polynomial surfaces

Consider two real polynomials in three variables, defined on the 3-sphere, $S^3$. Is there some Bezout-type theorem, relating the intersection of two closed surfaces defined by these polynomials and ...
3
votes
0answers
123 views

Characterization of closed immersions at the level of perfect complexes

Let $f : X \to Y$ be a morphism of quasi-compact quasi-separated schemes. Is there a necessary and sufficient condition on the inverse image functor $\DeclareMathOperator{Perf}{\mathrm{Perf}} f^* : ...
1
vote
1answer
125 views

A weak analytic version of the valuative criterion of properness

EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that (a) $f$ is injective on points; (b) $f$ is local imbedding near each point $x\in ...
-4
votes
0answers
58 views

The coordinate algebra of $F^*$ as an algebraic variety [closed]

How do you regard $F^*$ as an algebraic variety. How can we show that its coordinate algebra is the Laurent polynomial ring.
3
votes
1answer
218 views

What if the base change of an algebraic space is representable

Let $k\subset L$ be an extension of fields of characteristic zero. Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme. I am sure there are ...
0
votes
0answers
98 views

Finite resolution by sums of line bundles on toric varieties

I hope I wasn't searching wrong keywords or overlooking some easy arguments to prove/disprove it. What I'm asking is the following: Let $X$ be a smooth complete toric variety. $\mathcal F$ a coherent ...
6
votes
2answers
522 views

Is being reduced a generic property of schemes?

(Naive formulation:) Let $X$ be an (irreducible) affine variety (over an algebraically closed field $k$) and $I$ be an ideal of the coordinate ring $R$ of $X$. Assume $Y = V(I)$ is equidimensional. ...
2
votes
0answers
69 views

Are quaternion algebras from Witt's theorem endomorphism rings of vector bundles?

Let $k$ be a field with char $k \neq 2$. For $a,b \in k^{\times}$, let $(a,b)$ denote the quaternion algebra with $i^2=a$ and $j^{2}=b$, and let $C(a,b)$ denote the projective plane conic given by ...
1
vote
0answers
83 views

Stack of curves and universal deformations

I've just started studying algebraic stacks and I have a very basic question. I've learned the notion of Deligne Mumford stack and I've seen as the stack of stable curves $\overline{\mathcal{M}_g}$ ...
1
vote
2answers
128 views

there exists a hypersurface H ⊂ X such that X \ H is Stein and L is trivial over X \ H

"Suppose that X is a compact projective manifold equipped with a K¨ahler metric ω. Let L be a holomorphic line bundle In general, there exists a hypersurface H ⊂ X such that X \ H is Stein and L is ...
0
votes
0answers
87 views

moduli space of meromorphic $G$-Higgs bundles

I want to clarify with some topics in moduli space of semistable $G$-Higgs bundles on curve $X$ (genus $g$ is large enough) of fixing topological type $d \in \pi_1(G)$. Simpson's construction gives us ...
7
votes
1answer
308 views

Is the functor of points of a scheme cofinally small?

Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full ...
3
votes
0answers
161 views
+50

Symmetric power of an etale map of curves

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th ...
7
votes
1answer
299 views

periods of Mixed Hodge Structures

Two Questions: First. As I know the notion of periods comes when one has two vector spaces over a subfield $k$ of $\mathbb{C}$ (usually given by two cohomology theories) and an isomorphism between ...
-5
votes
0answers
89 views

Why do we study numbers to the base 2,8,10 and not the numbers to the base 3,4,6,7,9? [closed]

Why do we study numbers to the base 2,8,10 and not the numbers to the base 3,4,6,7,9 ?
3
votes
1answer
215 views

Abelian varieties over $p$-adic fields

Theorem : Let $A$ be an abelian varieties of dimension $d$ over a field $k$, non-archimedian valued complete, i.e. $\mathbb{Q}_p$, then $A(k)$ contains a subgroup of finite index analytically ...
2
votes
1answer
282 views

on the local structure of schemes

Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into: ...
4
votes
1answer
247 views

Excellent schemes

I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
0
votes
0answers
113 views

How to find critical points of the following polynomial?

I am trying to find critical points of the following equation in $\mathbb{R}^n$: $$F(x)=d-\sum_i a_{i}x_i^2+H(x).$$ Here, $a_{i}$ are positive real numbers, and $H(x)$ is a $\mathbf{harmonic}$ ...
-1
votes
0answers
89 views

Example of a fibration into singular curves?

Is there an example of a variety endowed with a fibration into curves which are generically singular ?
5
votes
1answer
330 views

Some questions about the ring Z((x))

$\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\dim}{\text{dim }}$ Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...
4
votes
1answer
161 views

Jacobian of a semistable curve

My question is about the proof of Example 8 in section 9.2 of the book "Neron models." There we have a semistable curve $X$ over an algebraically closed field $K$ and we let $\pi\colon \widetilde{X} ...
6
votes
1answer
229 views

Why do we need localization by Leftschetz motive?

Definition of the Grothendieck group and Leftschetz motive. The Grothendieck group of varieties is a free abelian group generated by classes of algebraic varieties with the following relation: $$ ...
-2
votes
0answers
174 views

A property P of morphisms of $S$-schemes $f : X \rightarrow Y$ is local on $X$, or $Y$, or $S$ or [migrated]

I have asked the same question on math.stackexhange here, but thought that is was a good idea to post it here also. I am learning schemes theory at school and I have for now only lectures notes that ...
2
votes
1answer
187 views

twists of algebraic groups

If $k$ is some field - for convenience, of characteristic 0 -, $\bar{k}$ is an alg. closure of $k$, and $G$ is some $k$-algebraic group, one can define a twist of $G$ to be some $k$-algebraic group ...
0
votes
1answer
167 views

Hilbert polynomial for any invertible sheaf

Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal ...
0
votes
0answers
98 views

Is the natural map $Pic A[M] \rightarrow Pic A[N]$ injective?

Let $A$ be a commutative ring. Let $M\subseteq N$ be an extension of positive seminormal monoids. Is the natural map $Pic A[M] \rightarrow Pic A[N]$ injective?