All Questions
22,548 questions
0
votes
0
answers
152
views
Kählerdifferentials and normal crossing divisors
Let $k$ be an algebraically closed field of arbitrary characteristic, $X$ a smooth surface over $k$, and $D_i \subset X$ be an regular, effective Divisor such that $D=\sum D_i$
has normal crossings ...
4
votes
0
answers
495
views
Spectral sequence for cohomology of open subset
Let $X$ be a smooth compact orientable manifold (or variety) and let $j: U \subset X$ be the complement to a union $\bigcup_{i \in I} X_i$ of smooth compact orientable manifolds. Suppose that $A$ is a ...
1
vote
1
answer
174
views
$Hilb_{lines}^{x}(X)$ and $Hilb_{lines}^{x}(X_{red})$
Let $X$ be a irreducible closed subscheme of $\mathbb{P}^N_{\mathbb{C}}$,
and $U$ is a nonempty open where $X$ is smooth and moreover for every $x\in U$ and for every line $l\subseteq X$ with $x\in l$ ...
- 1,159
0
votes
1
answer
487
views
Log resolutions of linear series
Let $X$ be a complex normal projective variety, let $|L|$ be a non empty linear series on $X$ and let $b(|L|)$ be its base ideal.
Suppose $f:X'\rightarrow X$ is a log resolution of the ideal $b(|L|)$.
...
- 1,150
1
vote
1
answer
426
views
Normal bundle of a quotient map
I would like to know if there is a notion of normal bundle to a quotient map. In one specific case, let $G$ be a finite group acting on an algebraic variety $V$. Denote by the map $\pi:V \rightarrow V/...
- 690
0
votes
1
answer
334
views
What are the Compact Symmetric Kahler Algebraic Varieties?
Here are some direct questions at the interface of algebraic and differential geometry:
(1) Is there an easy characterisation of those affine algebraic varieties which are Kahler?
(2) Is there an ...
- 1,776
3
votes
1
answer
335
views
Decomposition of primes, where the residue field extensions are allowed to be inseparable
I've been dealing with the following situation:
Let $R\subseteq S$ be an extension of Dedekind rings, where $Quot(R)=:L \subseteq E:=Quot(S)$ is a $G$-Galois extension. Let $\mathfrak{p}$ be a prime ...
- 1,386
5
votes
0
answers
323
views
Vector bundles of schemes and their topological realizations
Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$.
Does $R_\mathbb{R}$ send an ...
- 103
2
votes
0
answers
271
views
When does the smoothing of projectivized tangent cone lift to a deformation of a space?
Let $(X,0)\subset(\mathbb{C}^N,0)$ be the (formal) germ of a singular space (isolated singularity). Let $\mathbb{P}T_{(X,0)}\subset\mathbb{P}^{N-1}$ be its projectivized tangent cone (considered as a ...
- 2,232
2
votes
0
answers
391
views
Boundary behavior of Kähler cone with curvature restriction
Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.
A fundamental result is due to Demailly and Paun: they ...
- 247
2
votes
1
answer
534
views
Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?
Suppose we are given a map $f: X \to Y$ between two Riemann Surfaces, with branch points $p_1,p_2,\dots,p_n$ and known multiplicities at these points. Assuming we have a basis of $H_1(Y, \mathbb{Z})$,...
- 77
1
vote
0
answers
83
views
lift sections on a thickened curve
Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X.
Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective ...
- 3,472
3
votes
0
answers
217
views
Extending intersection bundles
Let $X$ be the product $Gr_i(V)\times Gr_j(V)$ of two Grassmannians where $V$ is a complex vector space of dimension $d$. There is an open $U\subset X$ formed by all those $(V',V'')\in X$ such that $V'...
- 23.5k
2
votes
0
answers
136
views
Coherent systems on K3 surfaces
Does anyone know whether the theory of coherent systems on $K3$ surfaces has been studied and, if yes, can you give me a reference? In particular, is there an analogue of Gieseker stability and of ...
- 773
3
votes
0
answers
361
views
A presentation of a scheme as a limit of smooth ones over finitely generated bases
Suppose that a scheme $S$ is separated, excellent, and has finite Krull dimension. Which of the following statements are true:
If $S$ is regular, then it can be presented as a projective limit of ...
- 16.9k
4
votes
0
answers
324
views
are there any results about equation over rational field or the extension Q[x]?
Given a polynomial $p(x)\in \mathbb{Q}[x]$, it is known that its roots can be obtained in terms of the coefficients of the polynomial by formulas involving the usual algebraic operations (addition, ...
- 3,104
3
votes
0
answers
118
views
Extending cohomology classes to compactifications of Kuga varieties
I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon,
available at http://www.ams.org/journals/tran/1993-...
0
votes
0
answers
251
views
Does the normalization of a projective morphism determine the line bundle?
Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms
$$f:X \to \mathbb{P}^n$$
and
$$g:X \to \mathbb{P}^m,$$
such that the image of $f$ is the ...
- 1
1
vote
1
answer
124
views
Lifting infinitesimal deformations for coverings
Let $f:X \rightarrow Y$ be an (unramified) holomorpic covering map between two (maybe non compact) complex manifolds.
Q: Does every infinitesimal deformation of Y lift faithfully to an infinitesimal ...
- 11
2
votes
1
answer
271
views
behavior of places of a function field under automorphism
if $P_{1}$ and $P_{2}$ are distinct places of equal degree of the function field F/K, and $\sigma$ is a K-field automorphism, such that $\sigma(P_{1})=P_{2}$. then, does $\deg (P_{1}\cap K(x))=\deg (...
- 123
1
vote
0
answers
170
views
Definition of the $L^2$-metric for the Determinant of Cohomology of a Vector Bundle on a Riemann surface
I start describing my setup. $X$ is a Riemann surface with a metric which can have a finite number of singularities, $E$ is a vector bundle on $X$ equipped with an Hermitian structure.
In an article (...
3
votes
0
answers
239
views
properness of stack
Hi,
assume we have an algebraic stack $A$ over $Sch(\mathbb{Z})$ which is quasi-compact and with separated diagonal. Assume that I have a stack $B$ which is obtained by rigidifying $A$ by a subgroup ...
- 39
1
vote
1
answer
443
views
Sections and subgroups in a unipotent group
Let $U$ be a smooth connected unipotent algebraic group over an algebraic closure $k$ of $\mathbf{F}_p$. Let $U' := [U,U]$ be the derived (= commutator) subgroup, and assume it is central in $U$. Let $...
- 20.2k
12
votes
0
answers
530
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
3
votes
1
answer
297
views
Families of sheaves on arithmetic varieties
Given an arithmetic variety $f: X \rightarrow Spec(\mathbb{Z})$.
Is there a notion of boundedness for families of sheaves on $X$?
I only found the notion for families on the fibers of $f$. But i am ...
- 1,391
2
votes
1
answer
414
views
generators of the ideal of an unipotent-generated algebraic group
Given any affine algebraic group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$ with a faithfull representation in $GL_n(\mathbb{F})$ . If one knows the generators of the ...
- 53
0
votes
0
answers
309
views
Weight filtration of MHSs
This is probably a very stupid question, but could someone explain to me where the weight filtration of mixed Hodge structures come from and why we actually need it?
If the Hodge-to-de Rham spectral ...
- 191
1
vote
0
answers
116
views
A certain 'coniveau-like' filtration for cohomology: what can one say about the intersection of $Ker H^i(X)\to H^i(Z)$ for $Z$ running through subvarieties of $X$ of dimension $m$?
Let $X$ be a smooth variety (say, a complex one; denote its dimension by $n$). What can one say about the intersection of $Ker (H^i(X)\to H^i(Z))$ for $Z$ running through (closed, not necessarily ...
- 16.9k
2
votes
0
answers
256
views
Efficient computing critical points of algebraic function involved radical expression
I am interested in finding local optima of an algebraic function $f(X,Y)$. Suppose, that this expression involves radicals, for example $f(X,Y)= \frac{1}{2}(X+Y)-\sqrt{XY}$. The approach in which i am ...
- 696
2
votes
0
answers
164
views
Algorithms for "Ideals" in polynomial algebras over the max-plus semi-ring
I'm a beginner in tropical geometry, and I'm running into the following question:
In the usual polynomial ring over a field, one has algorithms (i.e. using a Groebner basis) for determining whether ...
- 1,509
0
votes
1
answer
175
views
An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 16.9k
12
votes
0
answers
582
views
Pencils with many completely decomposable fibers
Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree
in $\mathbb C^{n+1})$.
The fiber over $(\lambda:\mu) \in ...
- 8,828
3
votes
0
answers
145
views
Curves whose stable reductions do not contain rational curves
Let $X$ be a smooth projective curve over $K:=K(A)$. $A$ is a strict henselian ring, $A/m=k=\bar k$. Suppose $\cal X$ is a stable model of $X$, ${\cal X}_{s}$ is the special fiber.
My question is:
...
- 1,921
1
vote
1
answer
219
views
Quasi-coherent module given by modules and compatibility conditions in the language of commutative algebra
Short question
Can we describe a quasi-coherent module on a scheme by usual modules with respect to an affine cover, which satisfy some compatibility conditions, which can be formulated in the ...
- 63.1k
3
votes
0
answers
289
views
Terminal quasi-affine varieties?
Let $U$ be a normal, irreducible, quasi-affine variety over the algebraically
closed field $k$ and consider the ring $A = \mathscr{O}(U)$ of regular
functions on $U$. Write $Max(A)$ for the ...
- 31
1
vote
0
answers
243
views
Special properties of (the $\gamma$-filtration of) $K$-theory of affine varieties.
Let $A$ be a smooth affine variety of dimension $n$. Are there any facts known on $K_{\ast}(A)$ and its $\gamma$-filtration which do not hold for $K_*(V)$ for an arbitrary smooth $V$ (of the same ...
- 16.9k
4
votes
0
answers
179
views
Equivariant version of a spectral sequence in Beilinson-Ginzburg-Soergel
In Beilinson, Ginzburg, and Soergel, "Koszul Duality Patterns in Representation Theory" (comment 3.4), the authors outline a spectral sequence as follows:
Given a filtered complex algebraic variety $...
- 103
0
votes
0
answers
252
views
local hyper tor is zero
Hallo,
in a proof I read an argument which used the following "fact" which was no further explained:
if you have a closed point $x$ on a smooth projective variety over a field $X$ and denote with $k(...
- 623
2
votes
0
answers
546
views
Ring objects in the category of cocommutative coalgebras (aka Hopf rings).
I have recently been doing some calculations in topology which are naturally expressed in terms commutative ring objects in the category of cocommutative coalgebras. These have been studied for quite ...
- 4,990
7
votes
0
answers
433
views
Ever seen a ringed group?
A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
- 12.4k
0
votes
1
answer
321
views
Why the curve [t^4,t^3s,ts^3,s^4] is not projectively normal in P^3?
Hartshorne EX I 3.18 b
Define a curve by [t^4,t^3s,ts^3,s^4]. It is actually a P^1.
Why the curve [t^4,t^3s,ts^3,s^4] is not projectively normal in P^3?
- 3,804
5
votes
0
answers
338
views
Orbital integrals and cohomology of compactified Jacobians
I have heard it said that the backdrop to the Laumon-Ngo proof of the fundamental lemma was a series of reductions in which orbital integrals were replaced by their analogues over p-adic fields which ...
- 8,723
4
votes
0
answers
258
views
Augmented Stable Base Locus and Positive Definite Singular Hermitian Metrics
I'm a bit confused about a lemma that I came across in Demailly's lecture notes on hyperbolicity, and how it is related to the notion of the augmented stable base locus.
Namely, consider a big ...
- 187
2
votes
2
answers
218
views
Why is an absolute value generated by a simple subvariety of a variety V well-behaved?
I am reading "Fundamentals of Diophantine Geometry" by Serge Lang.
Let V be a (absolute) variety, W be a simple subvariety of V. Then we know that the local ring of W is a discrete valuation ring, ...
- 750
4
votes
0
answers
211
views
find whether a polynomial has a zero in a finitely generated subgroup of $C^*$
Let $\Gamma$ be a finitely generated
subgroup of $C^*$. For a polynomial
$P\in Z[x_1,...x_k]$, determine whether
$P(x_1,...x_k)=0$ has a zero in
$\Gamma$. Is this decidable?
Motivation is ...
- 1,299
5
votes
0
answers
288
views
Syzygies of the singular locus of a nodal plane curve
Let $C\subset \mathbb{P}^2$ be a reduced nodal complex plane curve of degree $d$. Let $\Sigma$ be the set of nodes of $C$, and let $I$ be the ideal of $\Sigma$. Denote with $S=\mathbb{C}[x,y,z]$ the ...
- 3,338
2
votes
2
answers
276
views
Obstruction for real subvariety to be embedded as complex subvariety
Let $X$ be a nonsingular complex projective variety. Suppose $X$ is embedded as a nonsingular real subvariety of complex projective space ${\mathbb{CP}}^n$.
When can we embed ${\mathbb{CP}}^n$ in ...
user2529
2
votes
0
answers
188
views
Is Pic( G((z)) ) = $\mathbb{Z}$?
There are a fair number of papers by Beauville, Laszlo, Sorger, Kumar and others on the geometry of $LG/L^+G = G((z))/G[[z]]$ where $G$ is a simply connected and simple group over $\mathbb{C}$. In ...
- 3,968
1
vote
0
answers
236
views
Can etale $X$-schemes be lifted to $Y$, where $X$ is closed in $Y$?
For a closed embedding (of varieties) $X\to Y$ let $U/X$ be etale. Is is true that there necessarily exists an etale $U'/Y$ such that $U'_X=U$? If this is wrong in general, are there any assumptions ...
- 16.9k
0
votes
1
answer
87
views
Question regarding closure of sets defined by the vanishing of holomorphic functions
Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g$...
- 3,245