All Questions
22,547 questions
0
votes
0
answers
169
views
Is degree a "strict -transform" birational invariant for surfaces in the complex projective 3-space?
(Edit) My question is as follows: My previous question was about a kind of "strict-transform birational" invariant not birational invariants as usual. So I just delete the question since it ...
1
vote
0
answers
80
views
smooth algebras and triviality of de Rham complex
Hi,
Let $R$ be a $\mathbb Q$-algebra and let $A$ be a smooth $R$-algebra. If $A$ is a polynomial algebra
$A = R[T_1,\dots,T_n]$, then it is easy to see that the natural map
$R \to \Omega^\bullet_{A/R}...
- 2,842
3
votes
0
answers
191
views
Self-intersections of trigonometric polynomials
Suppose that $p(t) = a_0 + a_1 \cos(t)+ b_1 \sin(t) + a_2 \cos(2t) + b_2 \sin(2t)$ is a quadratic trig polynomial with complex coefficients. Assume that $a_2$ and $b_2$ are not real multiples of the ...
- 203
0
votes
0
answers
283
views
Singular fibers of Kodaira dimension
I will be pleased if someone could give me some good reference to understand Singular fibers of Kodaira type. In fact I_6, I_3 and A_1. Thanks
- 645
1
vote
0
answers
361
views
Why projective flat with connected and reduced fibers are Stein morphisms?
Let $f:X\to S$ be a projective flat morphism with connected and reduced geometric fibers. Why do one have that $O_S\to f_* O_X$ is an isomorphism?
- 411
1
vote
1
answer
167
views
Why is multiplication with a scalar no global morphism?
Given a smooth projective surface $S$ over an algebraically closed field, a sheaf rings or algebras $R$ on $S$ and a simple left $R$-module $M$, i.e. $Hom_R(M,M)=k$.Then we have $Hom_R(M,M(-i))=H^{0}(...
- 1,391
1
vote
1
answer
213
views
Constructing invertible sheaves out of actions of finite groups
Let $X$ be a quasiprojective algebraic variety over $\mathbb{C}$ with structure sheaf $\mathcal{O}_X$ and let $G$ be a finite group which acts freely on $X$. The quotient $Y = X/G$ is thus a ...
- 13
4
votes
0
answers
211
views
Dieudonne functor on truncated Barsotti-Tate groups
Let X/k be a smooth scheme over a perfect field of characteristic p. Is the Dieudonne functor inducing an equivalence of categories between truncated Barsotti-Tate groups and truncated Dieudonne ...
- 335
1
vote
0
answers
335
views
Universally open morphism with reduced fibers.
Hi.
I asked in the last post if, for a flat morphism $f:X\rightarrow S$ of complex spaces with reduced fibers and $S$ reduced, $X$ is reduced or not. In the algebraic setting, Liu said that the ...
- 435
2
votes
0
answers
214
views
Non-regular (Non-coherent) subdivisions of a polygon.
There are many papers and books which study about the regular subdivision of a convex lattice polytope.
My question is about "Non"-regular subdivisions of a 2-dimensional convex lattice polygon.
I ...
- 21
1
vote
1
answer
265
views
multiplicity under specialization
Let $C$ be a projective non-singular curve defined over a field $K$ with the characteristic zero. Let $y,z$ be non-constant rational functions defined $K$ such that $y$ is defined at all poles and ...
- 750
1
vote
0
answers
409
views
surjectivity of reduction for schemes smooth over Henselian base?
If $S$ is a Henselian local scheme with closed point $v$ and $X$ a smooth $S$-scheme, then it is well known that the canonical map $X(S) \rightarrow X(v)$ is surjective.
Suppose now that $Y$ is ...
- 646
4
votes
0
answers
215
views
cohomological dimension for coarser/finer topologies
Given a sheaf $\mathcal{F}$ with respect to some Grothendieck topology, is the cohomological dimension for this sheaf less than or equal to the cohomological dimension of a finer topology?
Example: $...
- 417
0
votes
0
answers
186
views
Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?
Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...
- 16.9k
4
votes
0
answers
202
views
The hypergeometric pullback conjecture
Here arXiv:math/0510287, Golishev proposed the following conjecture:
The hypergeometric pullback conjecture. Let $X$ be a Fano variety. Then,for any constituent $C$ of the quantum D-module $Q$ there ...
4
votes
1
answer
154
views
Is there a notion of "fibered category with boxproducts"?
Is there a notion of fibered category with box products?
By this I roughly mean a fibration $C\rightarrow B$ where $B$ has finite products,
along with functors
$$\boxtimes: C(X)\times C(Y)\rightarrow ...
- 13.2k
0
votes
1
answer
109
views
Action of SL(V) on P(V) and on vector bundles
Hi, i have some questions:
how $SL(V)$ acts on the projective line $\mathbb{P}(V)$ where $V$ is a k - vector spaces of dimension 2. And the same question for action of $SL(V)$ on structure sheaf $O, ...
- 1
0
votes
0
answers
131
views
Equalizers for morphisms of connected varieties with marked points
I recently began to study some aspects of varieties with marked points, and I tried to understand their categorical collective behaviors.
Here is one issue that I was not quite able to see ...
- 937
1
vote
0
answers
78
views
Question about $\Delta(n)_{U}$ notaion in illusie's cotangent compelexe et deformations
In illusie's book cotangent complexe et deformations, 38page, the notation $\Delta(n)_{U}$ appears, and I cannot find the direct explanation or hint about meaning of this notation in this book.
I ...
- 95
0
votes
0
answers
272
views
A simple(possibly trivial) question about Grothendieck Topologies
Let $C$ be the category of sets. Define coverings {$U_i\to U$} to be jointly surjective maps, i.e. $U$ is the union of the images of $U_i$. Then if $F$ is a sheaf of sets on $C$, is it clear that $F(\...
- 1,563
1
vote
0
answers
200
views
signature of $Pic(X)$
Let $X$ be a $K3$ surface and $\sigma$ an antisymplectic involution on $X$ and so $X$ is algebraic. 1.Why the signature of $Pic(X)$ is $(1,\rho -1)$? (This is well known but I cant find any direct ...
- 645
1
vote
0
answers
67
views
Closure of Petri locus
Let $X$ be an stable genus four curve obtained by gluing a hyprelliptic genus three curve at two points which are not hyperelliptic involution of each other. This curve corresponds to a point on the ...
- 51
1
vote
0
answers
101
views
How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.9k
3
votes
0
answers
171
views
negative even coefficients of virtual Poincare polynomials, with odd coefficients vanishing
Let $X$ be any complex projective (not necessarily smooth) variety. Let $P(X)$ be the virtual Poincare polynomial of $X$ defined by Deligne's mixed Hodge theory, that is,
$$P(X)=\sum_{i,j} (-1)^{i+j} \...
4
votes
0
answers
90
views
Homogeneity degree one functions of a matrix argument
I am interested in homogeneity degree one (scalar-valued) functions of a matrix argument. The simplest setup is as follows. Let $X$ be a symmetric $3\times 3$ matrix with real entries. Let $f$ be a ...
- 223
2
votes
0
answers
378
views
Quasi-projective orbifolds and algebraic line bundles
The notion of quasi-projective orbifold is generally accepted to contain at least the following: let $X$ be a (simply-connected) complex manifold, $G$ a group acting on $X$ by biholomorphisms, and ...
- 18.2k
4
votes
0
answers
99
views
Fibers of the secant map
Let $X\subset CP^N$ be a homogeneous (or perhaps just smooth) complex
subvariety and let $S^r(X)$ denote its abstract $r$-th secant variety
(the incidence variety in $X\times \cdots \times X\times CP^...
- 835
3
votes
0
answers
251
views
Orbits of semi-algebraic actions
Hello all,
I recently came across the following Theorem in Gibson (Singular points of smooth mappings, 1979). Since I haven't seen this result somewhere else and this reference is not so widespread, ...
- 461
1
vote
1
answer
276
views
weakly defective variety
Could anyone give us some references on weakly defective varieties?
Could anyone explain some good methods to show one variety is not weakly defective?
Thank you!
- 11
2
votes
1
answer
249
views
subspace topology for functors
let $X : Ring \to Set$ be a functor (a Z-functor in the language of demazure, gabriel) and $V \subseteq X$ a locally closed subfunctor. assume that $U \subseteq V$ is an open subfunctor. does then ...
- 63.1k
2
votes
1
answer
244
views
Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$
The blowdown of the zero section of the canonical bundle of the first del Pezzo surface $dP_{1}$, the blowup of $CP^{2}$ at one point, is a Calabi-Yau cone. I was just wondering if this cone admitted ...
- 169
0
votes
0
answers
70
views
Geometric effects of removing elements of D2n generalizable?
So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a ...
3
votes
0
answers
409
views
How looks the "land of Tamagawa numbers"?
Jonah Sinick's question here, other interesting ideas he mentioned, and Franz Lemmermeyer's remark make one think at Bloch and Kato's drawing + question. What's known or guessed about that "land" by ...
- 10.8k
3
votes
0
answers
209
views
Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor
Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
- 607
6
votes
0
answers
532
views
Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?
So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and ...
- 44.7k
2
votes
0
answers
106
views
What does the term "3-fold vertex" mean in enumerative geometry?
I read about enumerative geometry recently, namely something about the Gromov-Witten, Donaldson-Thomas and Pandharipande-Thomas invariants, and I was trying to see the picture.
It seems like the term ...
- 3,806
4
votes
0
answers
367
views
criteria for reduced fibres
I was wondering if it is foolish to ask if there is a criteria on a morphism $f: X \to Y$ between separated schemes of finite type over a perfect field which will assure that all the scheme theoretic ...
- 1,347
0
votes
0
answers
81
views
Another fibration with a given singular fiber class.
Let $f:X\rightarrow C$ be a fibration of complex manifold over a smooth curve $C$. Let $D$ be an irreducible component of singular fibers. How can one prove that $X$ does not admit any fibration with ...
- 1
1
vote
0
answers
192
views
Holomorphic vector fields with growth conditions on $X_\mathrm{reg}$
Let $M$ be a complex manifold with a hermitian metric (volumes and distances will be wrt this metric). Let $X\subset M$ be a complex analytic subspace of $M$ and $Y\subset X$ an analytic set ...
- 1,205
0
votes
1
answer
262
views
Subtleties in the construction of base change morphisms
Given a flat and projective morphism of noetherian schemes, $f: X \rightarrow Y$ and $F$, $G$ two coherent $O_X$-modules, flat over $Y$. Furthermore given a morphism $u: Y' \rightarrow Y$ of ...
- 1,391
4
votes
1
answer
358
views
Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).
Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$.
If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...
- 255
1
vote
0
answers
137
views
equality of intersection numbers
Consider the following situation: $X/k$ is a smooth projective geometrically connected curve with Albanese $A$ and Abel-Jacobi map $AJ: X \to A$, and $B/k$ an abelian variety.
Let $\alpha \in \mathrm{...
user19475
2
votes
0
answers
263
views
Literature Request: Genus Two Partition Functions
Apologies in advance for what will surely sound like a, "well why can't you just Google it" question, but I'm struggling to find good literature that presents the basic construction of genus two ...
- 21
2
votes
1
answer
406
views
Are there any criteria for a presheaf which is an etale sheaf to be a sheaf in the fppf topology?
I am happy to hear answers to variants too. For instance, my situation I actually have a sheaf in the smooth topology.
- 10.5k
1
vote
0
answers
215
views
Zeros of modular forms in higher dimensional cases
I have seen some results about the distribution and in particular number of zeros (in a fundamental domain) of a modular form on the upper half plane. Are there similar results about
modular forms ...
- 395
1
vote
0
answers
356
views
Quadratic Solutions
There are quadratic solutions to $x^4+y^4 = z^4$ in $\mathbb{Q} (\sqrt{-7})$. But for equations such as $x^4+y^4 = nz^4$ where $n \in \mathbb{N}, \ n \neq 1$ do there still exist extension fields of $\...
- 1
1
vote
0
answers
210
views
Prime Fano manifolds of coindex 4 or 5.
Let $X\subset\mathbb{P}^N$ be a non-degenerate smooth complex projective variety (manifold for short).
Recall that $X$ is called prime Fano if $Pic(X)=\mathbb{Z}\langle \mathcal{O}(1)\rangle$ and if ...
- 1,159
-3
votes
3
answers
400
views
Dense section of sheaves of modules
Here is something that isn't yet very clear to me. Say, I've got a commutative ring A. I consider the affine scheme from A, so it's a sheaf of rings over Spec A.
EDIT: And additionally let's say ...
- 2,275
1
vote
0
answers
330
views
Abel-Jacobi map for regular fibered surfaces.
Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let $...
- 11
2
votes
0
answers
127
views
Properties of special Cayley-Bacharach bundles on a K3-surface
Assume we have a $K3$-surface $X$ over $\mathbb{C}$ and two rational curves $C_1$ and $C_2$ on $X$ with $C_1.C_2=1$ and $C_i^2=-2$.
Let $x$ be a closed point on the reducible curve $C_1\cup C_2$. We ...
- 1,391