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

learn more… | top users | synonyms (1)

1
vote
0answers
39 views

The topology of Fano schemes of lines

Is there any references concerning the computation of the fundamental groups and Hodge numbers of Fano schemes of lines in a smooth hypersurface in $\mathbb{P}^n$?
0
votes
0answers
32 views

Is it possible to find an explicit definition of the “universal” (co)tangent bundle?

Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$ maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point $y \in \mathbb{P^1} $ ...
0
votes
2answers
77 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
1
vote
0answers
44 views

first chern class versus compactifying divisor in Ramanujam's surface

I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...
1
vote
1answer
82 views

Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...
1
vote
0answers
78 views

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ...
3
votes
0answers
263 views

Do those manifolds atrached to L-functions give rise naturally to motives? [on hold]

Edited after Will Sawin's comment: Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product ...
1
vote
1answer
87 views

Tangent cone of a complete intersection

Let $X$ be a quasi projective variety over $\mathbb{C}$. By the tangent cone of $X$ at a point $p \in X$, I mean the subvariety of the tangent space of $X$ at $p$ as it is defined in Harris' ...
2
votes
0answers
139 views

Is this diagram of sheaves actually Cartesian as claimed?

The question is about Corollary 1.6.2 (b) in the book by Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$ and morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ of sheaves on a ...
0
votes
0answers
60 views

What is the relation between the $K_0$ of a singular curve and its normalization?

Let $X$ be a singular curve over a field $k$. We define $K_0(X)$ to be the Grothendieck group of the category of coherent sheaves on $X$. For $X$ we have its normalization $\widetilde{X}$ and hence ...
4
votes
0answers
114 views

On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
6
votes
0answers
154 views

Are curves over imperfect fields defined over a smaller field?

Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
1
vote
0answers
73 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...
3
votes
0answers
138 views

Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family of degree $2$ maps defined (for $t$ small and non zero) by $$u_t([X,Y]) := [X^2, t Y^2, XY].$$ Note that as $t$ goes to zero, ...
1
vote
0answers
49 views

Does there exist a projection (of a variety) birational onto its image and satisfying additional conditions?

Let $X \subset \mathbb P^n$ be an irreducible (projective) variety of dimension $k < n-1$. By Harris [Har, Lecture 18, page 224], the projection $\pi_p : \mathbb P^n - \{p\} \to \mathbb P^{n-1}$ ...
4
votes
1answer
112 views

Descending a monomorphism of stacks

The question is about Proposition 3.8.1 in Laumon and Moret-Bailly book on algebraic stacks. Let $S$ be a scheme and let $F: \mathscr{X} \rightarrow \mathscr{Y}$ be a morphism of $S$-stacks (for the ...
-1
votes
0answers
109 views

Which classes of geometric results or problems can not be achieved by algebraic methods [on hold]

Algebra has been using when necessary as a tool to solve geometric and topological problems. I have seen algebraic results through geometric methods in the literature. Which classes of geometric and ...
2
votes
1answer
184 views

Which base change preserves integrality of schemes

Let $f:X \to Y$ be a flat morphism of projective noetherian integral schemes. Is there any known condition on a morphism $Z \to Y$ under which the resulting fiber product $X \times_Y Z$ is still ...
1
vote
0answers
110 views

algebraic closedness in in residue field [on hold]

If $A\subseteq B$ are affine domains over an algebraically closed field of $k$ of characteristic zero, such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
0
votes
1answer
121 views

irreducibility of general fiber

I would like to get a reference of the following fact. Let $A\subseteq B$ be affine domains over an algebraically closed field of characteristic zero. If $Q(A)$ is algebraically closed in $Q(B)$, ...
0
votes
0answers
47 views

Restriction of locally free sheaves and semi-stability on curves

Let $C$ be a stable curve and $\mathcal{F}$ be a locally free sheaf on $C$ such that the restriction of $\mathcal{F}$ to any of the irreducible component $C_i$ of $C$, $\mathcal{F}|_{C_i}$ is Gieseker ...
0
votes
0answers
29 views

the associated action on the transition functions

Let $X$ be a curve with an involution $\sigma$ generically unramified, given a $G-$bundle $E$ of rank $r$, than we ca take its pull-back, I want to describe the action of $\sigma$ on $G$. Fix a ...
0
votes
0answers
108 views

Is there any computation of $K^0(X)$ and $K_0(X)$ for a singular curve $X$?

Let $X$ be a projective curve over an algebraic closed field $k$ which characteristic zero. Define $K^0(X)$ as the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ as the Grothendieck ...
1
vote
0answers
72 views

Syzygies in integral domains

Let $f$ be a homogeneous irreducible element in a graded commutative Noetherian ring. What is the possible set of elements $f_1$ and $f_2$ such that $f=f_1g_1+ f_2g_2$? Even in very particular cases ...
0
votes
1answer
119 views

Gluing locally free sheaves on curves

Let $C$ be a quasi-projective curve, $C_i$ for $i=1,...,r$ are the irreducible components of $C$. Assume that $C_i$ is non-singular and $F_i$ locally free sheaf on $C_i$ of the same rank for all $i$. ...
-1
votes
0answers
71 views

Excercise of commutative rings, S. Balcerzyk and T. Józefiak [on hold]

Let $R=k[[x_1,...,x_n,y_1,...,y_n]]/(x_i y_j-x_j y_i)$, $i,j=1,\ldots,n$, where $k$ is a field. Prove that (a) $R$ is a domain. (b) $\dim R=n+1$. (c) $R$ is Cohen-Macaulay. (d) the type of $R$ ...
-1
votes
0answers
38 views

Cohen-Macaulay rings and Normal rings [migrated]

is there an example that R is Cohen-Macaulay but not normal ring? what about the converse example?
1
vote
1answer
83 views

Construction of curves and morphisms

Given any triple of positive integers $(g',g,d)$ with $2g'-2\geq d(2g-2)$. Does there always exist curves $C_{g'},C_g$ of genus $g',g$ with a degree $d$ morphism $f\colon C_{g'}\to C_g$? If we fix ...
6
votes
1answer
224 views

Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack? In many cases I ...
0
votes
0answers
28 views

Reflexive sheaves on stable curves-II [migrated]

This is an extension of Reflexive sheaves on stable curves. Let $C$ be a stable curve and $\mathcal{F}$ a reflexive sheaf on $C$ supported on the whole of $C$. Is the projective dimension of ...
0
votes
0answers
58 views

Reflexive sheaf on normal surfaces [migrated]

Let $X$ be a normal, projective scheme of pure dimension $2$ and $\mathcal{F}$ is a reflexive coherent sheaf on $X$. Is $\mathcal{F}$ locally free?
0
votes
0answers
62 views

induced map on tangent bundles from blow up morphism

Suppose $X$ is the plane nodal curve over $\mathbb{C}$. Then we can mimic what we do in differential geometry to define the "tangent bundle" over $X$ as a subvariety of ...
0
votes
1answer
123 views

Reflexive sheaves on stable curves

Let $C$ be a stable curve over an algebraically closed field of positive characteristic and $\mathcal{F}$ be a reflexive sheaf on $C$. Is $\mathcal{F}$ locally free? EDIT Is the projective dimension ...
0
votes
1answer
98 views

Base change of regular schemes [on hold]

Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X ...
3
votes
3answers
191 views

A question on flasque sheaf

Let $0\to \mathscr{F}'\to\mathscr{F}\to\mathscr{F}''\to 0$ be an exact sequene of sheaves. It is well known that $\mathscr{F}$ flasque iff $\mathscr{F}''$ flasque provided $\mathscr{F}'$ is flasque. ...
3
votes
1answer
420 views

learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig) Edit: You may assume knowledge of representation theory of finite groups ...
2
votes
0answers
207 views

A question about Weil restriction

Let $\pi:\tilde{C}\rightarrow C$ be a ramified cover between two smooth curves. And consider a group scheme $\mathcal G$ over $\tilde{C}$, I have found two definitions for Weil restriction: ...
0
votes
0answers
179 views

Is dimension invariant under blow-ups?

Let $X'\rightarrow X$ be a blow-up of a finitely dimensional scheme $X$ in a center $D$. Under which assumptions one has $\dim X'=\dim X$? Do you know a proof or a reference for a proof? Do you know ...
0
votes
2answers
141 views

étale cohomology via Cech cocycles for a quasi-projective scheme

I am looking for the explicit reference to the fact that for a quasi-projective scheme a class in the étale cohomology of a sheaf of a certain degree can by computed using Cech cocycles.
-2
votes
0answers
22 views

Find points which form angle X from a triangle formed by 2 other points [closed]

Here's a diagram (I can't post images) Diagram Given angle ACB, angle CAD, position C, and position E, and given that angle ADB equals angle ACB, find position D.
5
votes
1answer
123 views

Synthetic projective definition of cubic curves

In a synthetic (Pappian) projective plane, one can define a conic in various clever ways not referring to coordinates. For instance, if $f$ is a projectivity from the pencil of lines through a point ...
3
votes
0answers
93 views

Deformation of vector bundle on projective space with same Hilbert polynomial as multiple of structure sheaf

Let $E$ be a vector bundle on projective space ${\bf P}^n$ whose Hilbert polynomial is the same as $\mathcal{O}^{{\rm rank}(E)}$. Does there exist a vector bundle over ${\bf P}^n \times {\rm ...
2
votes
0answers
101 views

Hodge numbers of non-commutative varieties

Let $(X, \mathcal{A})$ be a non-commutative variety, by this I mean $X$ is a (smooth) algebraic variety and $\mathcal{A}$ is a sheaf of algebra on $X$. One such example in my mind is when $X$ admits ...
-1
votes
0answers
106 views

Non-flat fibration - 1. fibres still homotopic? 2. references/examples? [closed]

I stumbled over fibrations $\pi: E\rightarrow B$ that are not flat, i.e. where the dimension of the fibre $\pi^{-1}(b)$ jumps over certain points $b \in B$. Are these still 'ordinary' fibrations in ...
2
votes
1answer
130 views

Relative Proj and generation of sections

Let $π\colon Y = \mathrm{Proj}_B \mathcal{A} \rightarrow B$ be a morphism constructed from a coherent graded sheaf of $\mathcal{O}_B$-algebras $\mathcal{A} = \bigoplus_k \mathcal{A}$. I am looking ...
-1
votes
0answers
110 views

Groupes fondamentaux de Tate mixte [closed]

Have anyone ever read deligne and Goncharov's paper,Could you give me some idea why the formula(5.16.1) is true?this paper is very easy to find on the internet.So please forgive me not giving the ...
4
votes
0answers
132 views
+50

Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
13
votes
1answer
471 views

Must an algebraic variety with trivial tangent bundle be an abelian variety?

Suppose $X$ is a proper algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not ...
3
votes
0answers
128 views

Why write GRR with the relative tangent sheaf?

The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form $$ \operatorname{ch}(f_!\alpha).\operatorname{Td}(Y) = ...
3
votes
1answer
146 views

A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...