All Questions
22,547 questions
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Ext groups and flat modules on a K3 surface
Let $S$ be a $K3$ surface and $X$ the moduli space of some stable sheaves on it. Let $G$ be the universal family on $X\times S$ and $F$ the ideal of section of $X\times S\to X$. Knowing that for every ...
- 773
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votes
0
answers
159
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a question on the Poincar\'e bundle
Let $C$, a smooth curve. Let $J$ its Jacobian, consider the Poincar\'e bundle $\mathcal{P}$ on $J\times J$. Let $p: J\times J\rightarrow J$ the projection.
How can I compute the complex $R p_{*} \...
2
votes
0
answers
371
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the torsion part of grothendieck group
what is the geometric meaning of the torsion part of grothendieck group of bounded derived category of coherent sheaves and bounded derived category of R-module（R is a ring）？
- 165
3
votes
0
answers
356
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Colimit of an etale diagram of schemes
It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More ...
- 15.5k
2
votes
0
answers
242
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flat extensions of a locally trivial fiber bundle over punctured relative curve
Let $C$ be a curve (smooth projective over a field k) and $p$ a rational point on $C$. Put $\dot{C}=C-\{p\}$. Set $T=Spec R$ where $R$ is a noetherian k-algebra. Let $Y$ be a locally trivial fiber ...
- 51
4
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answers
152
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Is there an ellipsoid with given outer normals?
Pick two points $(x,0)$ and $(0,y)$ (say $x>0$ and $y>0$). Pick a unit vector $u = (u_1,u_2)$, $v = (v_1, v_2)$, and attach one to each of the points. Provided $u$ and $v$ are "nice" ($v$ needs ...
- 1,069
3
votes
0
answers
102
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Versions of Helly's Theorem for Unbounded Parallelpipeds
I'm studying Frobenius splitting of toric varieties based on Sam Payne's characterization of Frobenius splitting ("Frobenius splittings of toric varieties" (2008)) and I came across a sort of Helly's ...
- 31
3
votes
1
answer
124
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Continuous sections of the morphism ${GL}_{n}(A) \to {GL}_{n}(A/I)$, where A is a topological ring and I denotes a nilpotent ideal.
For which topological rings $A$ does there exist a continuous section (as a set map at least) of the quotient morphism $GL_n(A) \to GL_n(A/I)$, where $I$ denotes a nilpotent ideal in $A$?
It should ...
- 91
2
votes
1
answer
173
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Projective Curves which are Principal Bundles
I have a very specific question: does anyone know of a (non-trivial) example of a projective curve which is also a homogenous space (or just a principal bundle)? The trivial example being CP^1 = SU(2)/...
- 1,462
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votes
0
answers
167
views
Canonical bundle and diagonal morphism on varieties
Hi,
this question concerns the canonical bundle $\omega_X$ on a smooth projective $k-$variety $X$.
I want to consider the diagonal embedding
$i:X\rightarrow X\times X$
on $X$.
Furthermore denote ...
- 183
3
votes
0
answers
72
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Are two "nice" transformation groupoids with the same coarse moduli and isomorphic inertia isomorphic?
Hi!
I am stuck with the following question: suppose we have a semisimple connected algebraic group acting on a quasi-affine variety X by closed orbits, and suppose that inertia is flat.
Suppose we ...
- 31
0
votes
0
answers
100
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Is this set of curves discrete?
Let $\alpha_1, \dots, \alpha_n$ be complex numbers whose sum is zero and $u_1, \dots, u_{2g-2+n}$ be pariwise distinct nonzero complex numbers. Consider the the set of smooth genus g curves with n+1 ...
1
vote
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answers
70
views
reduced group covers of a curve
Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a ...
- 314
8
votes
0
answers
443
views
Lifting sections of bundles
Assume that $X$ is a quasiprojective scheme with a fixed imbedding into a smooth scheme M given by an ideal sheaf $\mathcal{I}$. Assume further that there is a locally free sheaf $E_X$ on $X$ that is ...
- 3,917
1
vote
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answers
149
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Morphisms, inducing isomorphisms on global functions
Let $f:X\to Y$ be a surjective morphism of complex varieties with $Y$ affine. Assume that every fiber of this morphism has the property that all the global functions are constant.
What else do I need ...
- 1,590
3
votes
0
answers
203
views
To what extent do results for smooth varieties hold for smooth Deligne-Mumford stacks?
I want to use some results true for non-singular varieties on smooth Deligne-Mumford stacks.
1 -- Concrete example: equivariant pullback for cohomology and K-theory of singular spaces that we could ...
- 31
1
vote
1
answer
251
views
Why is the Brauer Loop Scheme Not a Variety?
I am trying to grapple with the basics of scheme theory. Is the scheme defined by Spec[C[x,y,z]/(xy,yx,zx)] a variety? What do the points look like?
I suspect it represents points satisfying xy = ...
- 22.8k
4
votes
0
answers
172
views
Tangent cones to Severi strata
Let $\mathbb{C}[[x,y]]/f(x,y)$ be a reduced plane curve singularity. The base of a versal family can be taken to be (an open subset in) $\Lambda = \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$; the ...
- 8,723
-1
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1
answer
265
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About sufficient condition for smoothness
Dear Brian.
The dimension 1 case is very special. We assume $X$ no compact (Stein if we want), normal and of dimension >1...
In fact, i want to prove the following:
Let $f:X\rightarrow S$ be an ...
- 435
1
vote
0
answers
186
views
Quotient category $Coh(X)_d/Coh(X)_{d-1}$
Let $X$ be a regular scheme, and $Coh(X)_d$ be the category of coherent sheaves of $\leq d$ dimensional support.
Why is $Coh(X)_d/Coh(X)_{d-1}$ equivalent to $\bigoplus_{x \in X_d} \mathcal{A}(\...
- 417
3
votes
0
answers
240
views
Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?
(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question).
Consider the formal plane $\operatorname{Spec}...
- 44.7k
3
votes
0
answers
76
views
About the MacMahon functions on dimensions bigger than Three
As is well known, the generating functions of partitions counting in dimension 2 and 3 admit closed formulas. These functions show up in Euler characterisitics of Hilbert scheme of points in surfaces ...
- 399
1
vote
1
answer
248
views
Unusual ray tracing
Background
Ray tracing is very common in computational geometry and the problem is then to find the point of intersection between the equation of a line and the equation of a plane in 3D.
The ...
- 13
2
votes
0
answers
396
views
Noether-Lefschetz locus in enumerative geometry.
It is well known that if you have a smooth quartic surface $X\subset \mathbb{P}^3$, it may or may not have lines in it. Indeed, $X$ has the following options, 64 (the maximal number), 32, 16, or none.
...
- 2,132
2
votes
0
answers
290
views
quasi-trigonal curves
I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...
- 3,779
6
votes
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answers
267
views
Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
4
votes
0
answers
314
views
K-theory of non-reduced schemes
Suppose $X$ is a smooth, projective variety. Then Bloch's formula states that
$$CH^n(X)\cong H^n(X, \mathcal{K}_{n,X})$$
where $\mathcal{K}_{n,X}$ is the sheaf associated to the presheaf $U=Spec{A} \...
- 145
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votes
0
answers
101
views
Conditions under which the degree of an algebraic surface equals the degree of its planar section
An algebraic surface $\Phi$ in complex projective 3-space contains a circle $c$ such that the complex projective plane $P$ of $c$ intersects $\Phi$ only at the points of $c$. Assume that
$c$ is not a ...
- 1,488
1
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answers
161
views
Is -(E,E) greater or equal to 2 for a minimal resolution
I'm quite confused by the terminology minimal resolution and minimal model.
Let $f:X\longrightarrow Y$ be a minimal resolution of singularities, where $Y$ is a normal surface.
Let $E$ be an ...
- 73
2
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0
answers
129
views
Cuspidal stable curves
I have seen this many times but never know a rigorous proof. In a flat family of stable curves, if an elliptic tail is contracted then we get a cuspidal curve, if an elliptic
bridge is contracted, we ...
- 147
1
vote
1
answer
199
views
Closed image of a product of morphisms
This may be pretty trivial, but I can't figure it out. Suppose that $S$ is any scheme, and $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ are two morphisms of $S$-schemes, such that the closed image of each ...
- 13
2
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0
answers
220
views
When inverse image is conservative; a reference or a generalization?
I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...
- 16.9k
3
votes
0
answers
242
views
When is a category of groupoid schemes fibred over schemes?
The category of topological categories $Cat(Top)$ is fibred over $Top$ - the functor sending a groupoid $X_1 \rightrightarrows X_0$ to its object space $X_0$ is a Grothendieck fibration. Now one can ...
- 35.5k
3
votes
0
answers
259
views
Quotient of manifolds by groups and embeddings
Let $f:X_1\to X_2$ be a closed submanifold. Let $\rho:G_1\to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $\rho$-equivariant. I would like to get a ...
- 411
1
vote
2
answers
265
views
Is the smooth locus of a Q-Fano 3-fold rationally connected?
For me, the definition of $Q$-Fano $3$-fold is a normal $Q$-factorial projective $3$-fold with terminal singularities and $-K_X$ ample. If it is in addition Gorenstein, then this is true since it is ...
- 31
1
vote
0
answers
382
views
Is there functorial point of view to differential operator?
This question is related to differential operator in noncommutative geometry. I wonder whether there is any approach to differential operator that taking differential operator as a functor? I think it ...
- 1,305
3
votes
0
answers
238
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Invariant Subvarieties of Variety of Quiver Representations
I'd like to understand a special case of the following rather general algebraic geometry question:
Given an algebraic group $G$ acting on a variety $V$, can we describe the $G$-invariant subvarieties ...
- 31
1
vote
1
answer
183
views
complete ring as union of finite type algebras
Hi,
why the completion of a local ring $R$ can be written as an increasing union of $R$-algebras of finite type?
- 141
1
vote
0
answers
180
views
Generalized vector bundles with singularities on Riemann surfaces
Let $X$ be a Riemann surface of genus $g \geq 2$ or in other words a complex curve.
Let $P_1, \ldots, P_m$ be points in $X$ and $E \rightarrow X$ surjective map such that is is a complex $n$-...
- 121
1
vote
0
answers
98
views
Explicit operations with correspondences
Let $X: y^2=f(x)$ be an hyperelliptic curve over a finite field $k$. Consider two non-fibral correspondences $C,D\subset X\times X$. It is well known that they induce endomorphisms $\phi_C,\phi_D:...
- 293
3
votes
0
answers
120
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Codimension of $\text{Sing}(\overline{\mathcal{K}}_g)$ in $\overline{\mathcal{K}}_g$
Hi everybody,
let $\mathcal{P}_g$ be the moduli stack parametrizing pairs $(S,C)$ where $S$ is a K3 surface with a primitive polarization $L$ of genus $g$ and $C \in |L|$ is a smooth curve of genus $...
- 135
1
vote
0
answers
218
views
1-Parameter subgroups
Let $G$ be an affine reductive group defined over a field of characteristic zero $k$, denote by $\bar{k}$ the algebraic closure of $k$, and by $k^{\times}$ the multiplicative group of $k$. Let $Z(G)$ ...
- 143
6
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379
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ring-valued points of locally ringed spaces
of course, one should expect that the concept of ring-valued points is not well-behaved for locally ringed spaces (LRS). I want to see examples for this.
so consider $LRS \to Set^{Ring}, X \mapsto X(-...
- 63.1k
7
votes
0
answers
325
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Algebra A with Spec(A) reduced and Rep_n(A) non-reduced
As always, corrections to my misconceptions/misstatements are appreciated. This question is related to the following one, but in this question the algebras considered are commutative: Non-smooth ...
- 2,869
3
votes
0
answers
325
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Obstructions for reduced embedded deformation of Artinian rings
Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$...
- 30.6k
1
vote
0
answers
238
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relative flatness and torsion freeness
Hi.
Question 1: let $f:X\rightarrow S$ be a proper and surjective morphism of complex reduced spaces with $X$ pure dimensional. Let $F$ be a $S$-flat coherent sheaf on $X$. Is it true that the two ...
- 435
4
votes
1
answer
321
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Reverse Langlands transform
What os the meaning of a reverse Langlands transform to which Drinfeld seems to refer?
- 15.1k
0
votes
0
answers
115
views
the number of fixed points in geometric correspondance
Let $k$ be a finete field, $\bar{k}$ is an algebraic clousre of $k$, $\sigma \in Gal(\bar{k}/k)$ is the geometric Frobenius. Let $f:Y \to Spec(k)$ be a smooth, separated $k$-scheme of finite type, $...
- 1
1
vote
0
answers
270
views
References for shimura curve moduli of abelian varieties of dimension 3?
I have not much background of it ,so I want to konw is there any pepers study family of abelian threefolds parametric by shimura curve?
- 709
0
votes
0
answers
146
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$\mathbb{P}^1 \times \mathbb{P}^1$ bundles and genus 2 curve fibrations
I've been constructing genus 2 curve fibrations by starting with a (3,2) curve in $\mathbb{P}^1 \times \mathbb{P^1}$ and then promoting its scalar coefficients to sections of line bundles over some ...
- 175