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

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-1
votes
0answers
96 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 ...
0
votes
0answers
86 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
41 views

algebraic closedness in in residue field [on hold]

If $A\subseteq B$ are affine doamins 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
97 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
36 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
26 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
91 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
64 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
113 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 $i$. ...
-2
votes
0answers
68 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
79 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
217 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
53 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
121 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
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 ...
3
votes
3answers
187 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
votes
1answer
401 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
206 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
167 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
136 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 [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.
4
votes
1answer
118 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
85 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
99 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
104 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
129 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
109 views

Groupes fondamentaux de Tate mixte [on hold]

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 ...
3
votes
0answers
110 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
465 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
118 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
142 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
votes
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 ...
10
votes
1answer
302 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 ...
0
votes
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 ...
1
vote
0answers
232 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
votes
1answer
113 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
votes
2answers
230 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
vote
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 ...
1
vote
0answers
58 views

Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
2
votes
0answers
80 views

Dominating affine varieties over $k$ with affine smooth varieties over $k$

Given a geometrically integral affine variety $X:=\mathrm{Spec}(K[X_1,\ldots, X_n])/(f_1,\ldots, f_m)$ over a possibly imperfect field $K$, does there always exist an affine variety $\tilde{X}$ ...
7
votes
3answers
306 views

Exact sequences of groups and Tannakian formalism

By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered ...
0
votes
1answer
282 views

Reference for a lemma on étale maps

The Stacks Project has the following really nice Lemma concerning étale maps of rings: Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation $$ ...
6
votes
1answer
244 views

Acyclicity of the sheaf of real analytic differential forms

Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that the Poincare lemma holds for the de Rham ...
1
vote
0answers
65 views

Restriction of motivic nearby cycles

Let $h:Y\to \mathbb C$ be a regular map and let $f:X\to \mathbb C$ be the restriction of $h$ to a closed subvariety $X\subset Y$. Both $X$ and $Y$ are assumed to be smooth. The maps $h,f$ induce ...
1
vote
2answers
211 views

Ideals generated by two elements in the polynomial ring of two variables over a field

Let $k$ be a field. For example, $k=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime. Let $k[x,y]$ be the polynomial ring. Let $f,g\in k[x,y]$. Let the ideal $I=(f,g)$ be the ideal of $k[x,y]$ ...
2
votes
1answer
175 views

Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?

This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce ...
0
votes
0answers
88 views

Locally free sheaves and flat families of projective scheme

Let $f:X \to Y$ be a flat proper morphism of noetherian projective schemes and $\mathcal{F}$ is a coherent sheaf on $X$. Suppose for all $y \in Y$, $\mathcal{F} \otimes_{\mathcal{O}_Y} \mathcal{O}_y$ ...