All Questions
22,547 questions
0
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115
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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
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
2
votes
0
answers
134
views
Infinitesimal lifting for hensel schemes?
I have local hensel ring $A$ and a finite flat $A$-algebra $B$ (which is therefore a direct product of local henselien $A$-algebras) and I would like a section of the canonical map $B \to B / N$ where ...
- 1,347
1
vote
1
answer
183
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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
4
votes
1
answer
310
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Solvable subgroups of groups of polynomial automorphisms
Does every finitely generated free solvable group embed into the group of polynomial automorphisms of some C^n?
- 41
3
votes
0
answers
169
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Is there some short formula for the "defect" of Hilbert function
Let $X\subset\Bbb P^n_{\Bbb C}$ be a connected, Cohen Macaulay sub-scheme. (Possibly singular, reducible or non-reduced.) For $k\gg0$ the numbers $h^0(\mathcal{O}_X(k))$ depend polynomially on $k$. ...
- 2,232
6
votes
0
answers
238
views
Moduli space of modules with fixed length
Let $R$ be a (commutative) local Artinian ring, with an algebraically closed residue field $k$. I am interested in the set $L_n(R)$ of isomorphism classes of $R$-modules of length $n$.
If $R$ is a $k$...
- 30.6k
1
vote
0
answers
105
views
Secancy conditions for principally polarized abelian varieties
I. Krichever proved Welter's trisecant conjecture that says that a principally polarized abelian variety (over the complex numbers) is the Jacobian of some curve if and only if its Kummer variety has ...
- 663
1
vote
2
answers
218
views
Sheaf isomorphism.
Suppose you have a curve $C$ such that deg$K_C =0$ and $\Gamma(C,\Omega_C^1) \neq 0$. Does this automatically imply that $\vartheta_C \equiv \Omega_C^1$? My thought is yes, I've seen a proposition (...
- 13
1
vote
0
answers
194
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descent problem
Hi,
many times on books I find the following phrase: "...by (étale) descent we can reduce to the case..." (étale can be replaced by other topologies). What does really means and why "they" can reduce ...
- 11
1
vote
1
answer
219
views
Name for a module with only one associated prime
In EGA IV2, Def. 3.2.4, Grothendieck defines a quasicoherent sheaf over a locally Noetherian scheme to be "irredondant" if it has a unique associated point. Presumeably, a module over a Noetherian ...
- 7,338
6
votes
0
answers
229
views
Vanishing theorem for sheaves of logarithmic forms
Let $X=\mathbb{A}_2\cup B$ be a smooth complex projective completion of the affine plane $\mathbb{A}_2$ with boundary $B$ a simple normal crossing divisor with rational components (e.g $X$ a smooth ...
- 61
2
votes
0
answers
264
views
Tor group of the ideal sheaves of points on a surface
Let $S$ be an algebraic surface and $P,Q$ two points on it. Let $I_P$ and $I_Q$ denote the ideal sheaves of $P$ and $Q$ respectively. Which is the tor group $Tor^1(I_P,I_Q)$? And what about $Tor^1(I_P,...
- 773
1
vote
1
answer
316
views
intersection cohomology when the resolution is not semi-small
When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection ...
- 2,216
2
votes
0
answers
163
views
Is there an example of index 1 picardnumber 1 Fano 4-fold with h^{21}\neq0
Are there any known examples of index 1 smooth Fano 4-folds X with $\mathsf{Pic}(X)\cong\mathbb{Z}$ with $h^2(\Omega^1_X)\neq0$?
- 420
1
vote
0
answers
115
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singularities $\mathcal{A}_{g,d}$ in positive characteristic
Hi,
I would like to understand the singularities of the moduli spaces of abelian varieties with polarization of degree $d>1$ in characteristic $p>0$ when $p|d$. Do you know some good references?...
- 11
2
votes
0
answers
314
views
Tangent sheaf of $\mathbb{P}^1\times\mathbb{P}^1$
Hi guys!!I'm new in this forum. I have a simple question for you. Let $k$ an algebraically closed field. Consider $\mathbb{P}^1\times\mathbb{P}^1$ and $T_{\mathbb{P}^1\times\mathbb{P}^1}$ the tangent ...
- 21
1
vote
2
answers
120
views
Removing a hypersurface when applying the Representation theorem to prove Positivstellensatz with uniform denominators
Let $f$ and $g$ be positive definite forms in the polynomial ring ${\mathbb{R}}[x_0,\ldots, x_n]$ such that $\deg(g)$ divides $\deg(f)$. A generalization of a theorem by Reznick is that $g^N f$ is a ...
user2529
3
votes
0
answers
262
views
[Solved] Multiplicity of a Poincaré divisor at the points of order 2
Let $X$ an abelian variety /$k$, char($k$)=0, $k=\overline k$, and be $\widehat{X}$ its dual. With $P$ I will denote the (normalized) Poincaré bundle over $X\times_k\widehat X$. We have an action of $...
- 669
0
votes
0
answers
198
views
why a reduced ring can be embedded into a sum of integral rings?
Hi,
the question is exactly
"why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?"
Is this simply because in the normalization process we can have many irreducible ...
- 141
1
vote
0
answers
94
views
invariant lines avoiding fixed subvarieties
Could anybody help me with the following question ?
Assume we are given:
(1) a finite order (linear) automorphism $g$ of the complex projective space $\mathbb{P}^r$,
(2) a closed algebraic ...
- 71
0
votes
1
answer
113
views
Removing non-basepoint from linear sytem.
Can you tell me, why the following is true:
Let $C$ be a smooth, complete curve over an algebraically closed field. Let $D=P_1+...+P_n$ be an effective divisor that is linear combination of (not ...
- 101
1
vote
0
answers
268
views
Rational map defined over K leads to algebra question
Hello,
Concrete algebraic question : Let $K$ be a perfect field, $\bar{K}$ a fixed algebraic closure and let $f \in \bar{K}[x_1,\ldots,x_n]$. I was wondering when there exists another polynomial (non-...
- 11
2
votes
0
answers
136
views
Galois theory for real algebraic sets
Hi,
is there a "Galois" theory for rational functions of real algebraic sets
similiar to the one relating coverings of riemanien surfaces with Galois Extension of
meromofphic functions?
Thanks
- 949
1
vote
0
answers
194
views
Inverting infinitely many points on an algebraic curve
This question is very naive, but that's why I'm asking it.
Say we begin with $\mathbb{A}^1_{\mathbb{C}}$. Let $U$ be the open disc around $0$ of radius $1$. Now invert all the $a$'s not in $U$: $Spec(...
- 6,348
2
votes
0
answers
74
views
Weak admissibility in algebraic families
Let $M$ be an algebraic family of isocrystals over a base scheme $R/\mathbb{Q}_p$ (not a rigid analytic space).
The question is: is the set of weakly admissible points (i.e., the points $r\in R$ ...
- 1,841
1
vote
0
answers
178
views
$G_m$-cohomology of a motif (that corresponds to a stack?)
As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth $G$-...
- 16.9k
1
vote
0
answers
137
views
Smoothly patching together polynomial level sets
Apologies if this is too low-level; I wasn't sure whether it belonged here or on math.stackoverflow.
Say I have level sets for two polynomials of degree n on ℝd. What constraints must they satisfy to ...
- 3,612
4
votes
0
answers
293
views
degree of pull-back via F of an hyperplane vs degree of defining polynomials of F
Let X be a nice projective variety (say, smooth and defined over the complex numbers) embedded in some projective space, and let $F:X\to\mathbb{P}^n$ be a rational map given by $[f_0:f_1:\cdots: f_n]$ ...
- 41
7
votes
0
answers
295
views
Positivity properties of virtual Hodge numbers of Calabi-Yaus
Let $X$ be a normal, projective complex variety with an anticanonical divisor $D$. Do the virtual Hodge numbers of the noncompact Calabi-Yau variety $X$ \ $D$ enjoy some sort of positivity property?
...
- 27.9k
1
vote
0
answers
115
views
reading off invariants of a scheme $X$ from $D^b_c(X, \bar{\mathbf{Q}}_\ell)$
Which invariants of a scheme $X$ can be read off from $D^b_c(X, \bar{\mathbf{Q}}_\ell)$ (the bounded derived category of $\bar{\mathbf{Q}}_\ell$-sheaves on $X$, see e.g. [Kiehl-Weissauer])?
user19475
1
vote
0
answers
56
views
Combinatorical surface is restricted to a closed face an injection
Hello :)
I'm third year student of mathematics. In my own intrest i'm studying topology in combinatorical sense. Herefore i found also an lecture note in knot theory from Roberts. I want to understand ...
- 11
1
vote
0
answers
109
views
A good reduction property
Let $R$ be a normal noetherian domain and $K$ the quotient field of $R$. Let $X$ be a smooth algebraic $K$-scheme and $x\in X(K)$ a $K$-rational point of $X$. Does there exist a tripel $(U, s, i)$ ...
- 1,673
5
votes
0
answers
173
views
Maximal algebraic sub-groupoids
By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in the latter case are ...
- 35.5k
3
votes
0
answers
186
views
affineness vs geometric affineness
Let $X$ be $k$-scheme of finite type, $k$ being a (perfect) field, and assume $X\otimes\overline{k}$ is affine. Is $X$ necessarily an affine scheme? What about if $X$ is a $k$-group scheme?
- 4,265
2
votes
0
answers
234
views
Are the two B model constructions equivalent?
Now we have two B model constructions: Kontsevich-Baranikov (for genus 0) and Costello (for any genus). My question is, are they equal in genus 0? Or now which one is the one we want? Thanks!
- 1,499
5
votes
0
answers
181
views
Are $n$-vector bundles an $(\infty,n)$-symmetric monoidal category with duals?
In Lurie's On the Classification of Topological Field Theories, one of the main characters are $(\infty,n)$-symmetric monoidal category with duals. A basic example of this should be $n$-vector spaces, ...
- 6,777
3
votes
0
answers
135
views
Irreducibility of a representation of $\Gamma(N)$
Let $Y(N),$ for $N\ge3,$ be the modular curve over $\mathbb C$ with respect to the $\Gamma(N)$-level structure. Let $f:E\to Y(N)$ be the universal elliptic curve. Then $R^1f_*\mathbb Q$ is a ...
- 4,265
5
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0
answers
271
views
Any suggestion on the paper " Hodge cycles on abelian varieties" ?
I have to understand Deligne's paper" Hodge cycles on abelian variesties",but I find it very difficlut to read ,even though I have already spent several months to learn the etale cohomology.Is there ...
- 699
3
votes
0
answers
82
views
When can we discard higher order terms in a set of ideal generators?
Let $A=k[x_{1},...,x_{n}]$ be a polynomial ring over a field $k$, let $f_{1},...,f_{m} \in A$ be polynomials, and let $I=(f_{1},...,f_{m})$ be the ideal that they generate in $A$. Now suppose that we ...
- 1,329
4
votes
0
answers
118
views
Is a point that is incident to several circles not on the same sphere necessarily singular?
I have an irreducible polynomial $f \in R[x,y,z]$, and a point $p$ that is in the zero-set $Z$ of $f$. My question is, given the following properties of $p$, is it necessarily a singular point of $f$. ...
- 1,072
4
votes
0
answers
243
views
Chow groups of arithmetic surfaces
Given an arithmetic surface $S$, I would like to know the following properties of its first and second Chow groups $CH^1(S), CH^2(S)$:
Are they finitely generated? If so, what is the rank?
What is ...
- 4,593
2
votes
0
answers
227
views
Finding equations for projective bundles associated to vector bundles over explicitly given varieties
Suppose I have a projective variety V over a field k. It is given explicitly in terms of homogeneous equations. Moreover, say I have an explicitly given vector bundle E (in terms of a module ...
- 21
4
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0
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287
views
Rozansky-Witten class associated to the Theta Graph
Suppose I have a holomorphic symplectic manifold, and a smooth $(1,0)$ connection on the tangent bundle which is compatible with both the complex and the symplectic structures. Say that the ...
0
votes
1
answer
207
views
Dimension of H^0(S,O_{S}(-C))
Let $S$ be a smooth projective algebraic surface over $\mathbb C$ and $C$ be a smooth curve on $S$. Is it always true that $dim_{\mathbb C} H^0(S,O_{S}(-C))=0$ ? In particular, is it zero when $S$ is ...
3
votes
1
answer
184
views
How many linear terms are in the Hilbert set of H(z,t), a polynomial in 2 variables over a field k(s) of transcendence degree one over a finite field?
I am looking for a good reference for Hilbert's irreducibility theorem, and ofproperties of Hilbert sets besides Serres Lectures on The Mordell-Weil Theorem. In particular, I am interested it to the ...
6
votes
0
answers
236
views
Does an isomorphism between infinitesimal neighbourhoods induce a jet of a diffeomorphism?
Let $D$ be a divisor in a (compact) Kahler manifold $X$. Let $N$ be the total space of the normal bundle of $D$ in $X$ and identify $D$ with the zero section of $N$. Let $m\subset O_{X}$ and $p\subset ...
- 103
6
votes
0
answers
264
views
When do equivariant sheaves on a formal neighborhood extend?
Suppose that $X$ is a variety (in char 0) with an action of an affine algebraic group $G$. Let $Y \subset X$ be a subvariety fixed by $G$--the action map agrees with projection upon restriction to $Y$...
- 1,038
7
votes
0
answers
244
views
Projectivity of flops
Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $\mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $\widetilde{X}$ is Kahler, it ...
- 14.7k
2
votes
0
answers
254
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...