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

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1answer
26 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
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0answers
64 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 ...
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0answers
31 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
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72 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 ...
5
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0answers
123 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 ...
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0answers
55 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 ...
2
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0answers
93 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, ...
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0answers
43 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 $\pi_p : \mathbb P^n - \{p\} \to \mathbb P^{n-1}$ ...
3
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1answer
99 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 ...
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0answers
107 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
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1answer
179 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 ...
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0answers
105 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
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1answer
114 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 $Q(B)$, ...
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0answers
43 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 ...
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28 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 ...
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0answers
99 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 ...
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0answers
69 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 ...
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1answer
117 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 $i$. ...
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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 $R$ ...
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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
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1answer
81 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 ...
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1answer
221 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 ...
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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 ...
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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?
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59 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
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1answer
122 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
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1answer
96 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 ...
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3answers
190 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. ...
2
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1answer
412 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
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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: ...
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175 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 ...
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2answers
138 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.
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0answers
22 views

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

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
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1answer
120 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
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0answers
89 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
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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 ...
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0answers
106 views

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

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 ...
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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 ...
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0answers
123 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
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1answer
469 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 ...
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0answers
123 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
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1answer
144 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 ...
2
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1answer
168 views

'Stalk' of vanishing cycles at $k$-point

I have a simple question on notation. Let $S$ be a Henselian trait with closed point $s$ (with finite residue field $k$) and generic point $\eta$. Let $X/S$ be a variety. Then, we have the functor ...
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1answer
316 views

Is there a unique commutative group structure on $\mathbb{G}_m$?

Let $S$ be a scheme and let $X := \mathrm{Spec}(\mathscr{O}_S[t, t^{-1}])$ be the underlying $S$-scheme of the $S$-group scheme $(\mathbb{G}_m)_S$. Is there only one structure of a commutative ...
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0answers
94 views

Polynomial approximation on affine varieties [migrated]

Let $V,W \subseteq \mathbb{A}^n$ be two affine varieties over an algebraically closed field $k$ of characteristic zero and let $a,b\in k$. Q: Can we find a polynomial $f \in k[X_1,...,X_n]$ such ...
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233 views

Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$, $$\mathsf{F(p)}=a, ...
0
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1answer
114 views

Why does this vector bundle on the surface sit in this exact sequence?

Let $X$ be a K3 surface. Let $E$ be a semistable rank 3 vector bundle. Now suppose $0 = E_0\subset E_1\cdots\subset E_s=E$ be the Harder-Narasimhan filtration. Suppose $E_1$ is $\mu$-stable and rank ...
5
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2answers
232 views

Stabilisers of group actions

Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$. Suppose there exists some point $x \in X$ whose ...
1
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0answers
106 views

identity component of a formal group

Let $G=\operatorname{Spf} A$ be a formal group, the it is stated that the identity component $G^\circ$ (defined as $\operatorname{Spf} A_{\operatorname{fm}}$ for some open maximal ideal ...