All Questions
2,036 questions
2
votes
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answers
251
views
Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces
Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
- 1,509
2
votes
0
answers
415
views
relative cohomology $H(X,D)$ of a pair in Weil cohomology theory
In algebraic topology, one defines relative cohomology groups $H(X,A)$ of a pair of spaces $A\subset X$.
Is there an analogue in algebraic geometry of cohomology of a pair of schemes?
For example, ...
- 1,299
2
votes
1
answer
172
views
Global centralizers in Jordan-Chevalley decomposition in bad characteristic
Let $G$ be an affine algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra. Given $X\in\mathfrak{g}$, it has (...
- 53
2
votes
0
answers
285
views
infinite dimensional germs of schemes and tangent spaces
(The question of the type "how to define?")
Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...
- 2,232
2
votes
1
answer
230
views
curve through a point avoiding an hypersurface, II
Inspired by this question:
Suppose given an algebraic curve $C \subset \mathbb{A}^2$, and a point $x \in C$. Can you find another (closed) curve $D \subset \mathbb{A}^2$ such that $C \cap D = x$?
- 8,723
2
votes
1
answer
526
views
extending homomorphisms of Abelian schemes
Let $S$ be an integral scheme with function field $K = K(S)$. Let $\mathscr{A}, \mathscr{B}$ be Abelian schemes over $S$. Let $L/K$ be a separable field extension. Given $f_L \in \mathrm{Hom}(\mathscr{...
user19475
2
votes
0
answers
174
views
Interpretation of some maps involving cohomology groups
I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...
- 895
2
votes
1
answer
416
views
(Bridgeland stability conditions) How can I get the heart of a bounded t-structure on $D^b(P^3)$?
In the article, Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland stability conditions on threefolds. I Bogomolov-Gieseker type inequalities, J. Algebr. Geom. 23, No. 1, 117-163 (2014). ...
- 21
2
votes
2
answers
2k
views
Reference needed for representation theory of direct products of algebraic groups over a field (of arbitrary characteristic)
In my dissertation I proved a certain theorem(s) concerning the representation theory of a direct product G x H of algebraic groups over a field, given those of G and H. But I would wager 100:1 that ...
- 511
2
votes
0
answers
216
views
On an application of the going-down theorem of Cohen-Seidenberg in Mumford
There is a following result in Mumford's red book of schemes (Chapter II Section 8). Here $R$ is a valuation ring with algebraically closed fraction field $k$.
Let $Z \subset \mathbb{P}^n_k$ be an ...
- 3,625
2
votes
0
answers
293
views
Hypersurfaces in projective bundles over $\mathbb P^1$
I am working on a suggestion of a comment here.
Let $E \rightarrow \mathbb P^1$ be a non-trivial vector bundle of rank $r$ with $\deg E =0$ and $\mathbb P(E) \rightarrow \mathbb P^1$ be its ...
- 1,841
2
votes
0
answers
109
views
Shimura curves and quaternion orders without elements of norm -1
Let $O$ be an order of a quaternion algebra over $\mathbb{Q}$ such that $O$ does not contain elements of norm $-1$. Such orders exist, but seems less used, in particular these orders are not Eichler.
...
- 91
2
votes
1
answer
5k
views
Roots of bivariate polynomials
A bivariate polynomial of degree $m+n$ is,
$ p(x,y) = \sum_{k=1}^n\sum_{j=1}^m a_{jk}x^ky^j$
where $a_{mn}\neq0$ and $a_{jk}\in\mathbb{R}$ for $1\leq j\leq m$, $1\leq k\leq n$.
I would like to ...
- 3,217
2
votes
1
answer
187
views
Is every sufficiently general monic quartic rational square infinitely often?
Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...
- 25.4k
2
votes
0
answers
332
views
Killing cohomology of structure sheaf by pullback along Frobenius and finite etale covers
On a smooth projective variety $X$ over a finite field, you can pullback any element in $H^1(X,\mathcal{O}_X)$ by a combination of Frobenius and finite etale cover so it gets killed. In order to prove ...
- 5,901
2
votes
1
answer
86
views
Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$
The answer to this MO question says the following:
Lemma 1. Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified if and only if $h$ ...
- 2,837
2
votes
2
answers
627
views
Numerically negative exceptional divisor on a surface.
Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
- 2,215
2
votes
2
answers
461
views
Bound on the (anticanonical) degree of toric Fano varieties
Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ ...
- 13.5k
2
votes
1
answer
265
views
Question about automorphism functor in Sernesi's "Deformations of algebraic schemes"
Let $X$ denote an algebraic scheme over $\operatorname{Spec} k$ such that its deformation functor $\operatorname{Def}_X$ has a semi-universal couple $(R,u)$, where $R$ is an Artinian $k$-algebra and $...
- 749
2
votes
0
answers
144
views
Local freeness of $\pi_*F(r)$ from flatness of $F$
In 'Fundamental Algebraic Geometry' by Fantechi there is a lemma in section 5.3.2, page 119:
LEMMA 5.5 Let $S$ be a noetherian scheme and let $F$ be a coherent sheaf on $\mathbb{P}^n_S$. Suppose there ...
- 5,946
2
votes
1
answer
192
views
Relations between Dolbeault cohomology and the corresponding $L^2$-cohomology
Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}_{(2)}(X)$ respectively.
As is well known, on a compact complex manifold $X$, $H^{p,...
- 391
2
votes
2
answers
899
views
Number of generators of an ideal in a polynomial ring over a Noetherian ring
Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x_1, \ldots , x_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[...
- 545
2
votes
1
answer
398
views
Picard group of $\mathrm{GL}(n)$-orbits
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...
user168611
2
votes
1
answer
372
views
When is restriction an equivalence of categories of equivariant vector bundles?
Suppose a (linear algebraic) group $G$ acts on a variety $X$ and that $U$ is a $G$-invariant open subvariety. My question is: under what conditions is the restriction functor
$i^*: Vect^G(X) \...
- 853
2
votes
1
answer
476
views
Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph"
I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order ...
- 203
2
votes
2
answers
635
views
Are there other ways to show Pic(G)is trivial when G is a simple-connected semisimple algebraic groups over C?
Indeed,I'm reading the book《representation theory and complex geometry》,there is a proof of the fact that Pic(G)is trivial when G is a simple-connected semisimple algebraic group over C,but the proof ...
- 73
2
votes
0
answers
327
views
Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide
There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t) )$ be ...
- 1,168
2
votes
0
answers
1k
views
Extension by zero and quasi-coherence
The following makes probably sense for any site, but I stick for concreteness to the etale one.
Let $f: U \to X$ be an etale morphism. As explained, for example, in Remark 8.16 of Milne's Lecture ...
- 12.1k
2
votes
1
answer
1k
views
Line bundles, linear systems and normalization
One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on $\mathbb{P}^...
- 23
2
votes
3
answers
1k
views
Finiteness of étale Cohomology Groups
Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8):
Proposition: Let $F$ a constructible sheaf on $X_{et}$, the small étale site of $X$, $X$ proper over a ...
- 23
1
vote
1
answer
264
views
Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
Let $R \subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $...
- 2,837
1
vote
1
answer
218
views
stable sheaves in characteristic $0$
Let $K$ be a non-algebraically closed field of characteristic $0$ and $X_K$ a smooth, projective, geometrically connected curve defined over $K$. If $F$ is a stable locally free sheaf on $X_K$, is it ...
- 2,323
1
vote
1
answer
367
views
Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ($\...
- 413
1
vote
0
answers
147
views
Properties of pointless projective curves over finite fields?
Probably not research level, feel free to downvote.
We got construction of bounded degree projective curves
with no points over finite fields. This construction generalizes to higher dimension.
One of ...
- 25.4k
1
vote
0
answers
152
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
- 7,182
1
vote
0
answers
294
views
Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
1
vote
0
answers
98
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
1
vote
1
answer
333
views
Pushforward of semi-stable sheaves under finite field extension
Let $k$ be a field of characteristic zero and $X$ be a non-singular rationally connected variety over $k$. Let $L$ be a finite field extension of $k$. This induces a proper morphism $p:X_L \to X_k$. ...
- 1,981
1
vote
0
answers
133
views
Contractibility of a $K_0^{\oplus}$ presheaf
Let's assume $X$ is a smooth projective variety over a field. Let $\Delta^{\bullet}$ be the cosimplicial scheme over the same field, where at level $n$ is just the $n$-th algebraic simplex. We can ...
- 5,901
1
vote
0
answers
141
views
Homeomorphic endomorphism of schemes inducing equivalence of sheaves
Let $F: X \to X$ to be an endomorphism of scheme $X$, which is additionally assumed to induce an universal homeomorphism on the underlying topological space $| X|$. Then it is known that this induces ...
- 5,946
1
vote
0
answers
235
views
Récollement of stable $t$-structures
Given a recollement
$$
\mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} \...
- 13.6k
1
vote
1
answer
483
views
formally étale morphisms which are also universally closed
A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally ...
- 761
1
vote
2
answers
279
views
Correspondences acting on cohomology groups $H^*(X)$ & splittings
Let $X$ be a smooth connected proper scheme over field $k$. It is known that
correspondences $\alpha \subset X \times X$ regarded as
objects in Chow groups $\text{CH}^*(X \times X)$
act on cohomology $...
- 619
1
vote
1
answer
119
views
Multiplication on cubic hypersurfaces and partially defined groups
Let $V$ be a general smooth projective cubic hypersurface. Doing literally as in case of cubic curves we define a relation on $V\times V\times V$: $(x,y,z)$ satisfy it iff $x+y+z$ is an intersection ...
- 2,934
1
vote
2
answers
2k
views
What is the geometric point of view of an algebraic line bundle compared to a analytic line bundle?
Hi folkz,
I'm trying to learn more about line bundles, invertible sheaves and divisors on schemes. I understand the connection beweteen Cartier and Weil Divisors and the connection between Cartier ...
- 345
1
vote
2
answers
391
views
Orbits of rational functions
This is a generalization of Integrality of iterates of rational functions.
The question is: given an infinite sequence of rational numbers $a_0, a_1, \dotsc, a_n, \dotsc,$ is it always an orbit of $...
- 96.4k
1
vote
3
answers
999
views
On the Clifford index of a curve
Let X be an algebraic curve and c be the Clifford index of X.
When c is small (e.g c=1), what is the classification of the line bundle who computes c?
- 11
1
vote
1
answer
218
views
Dimension of Zariski closure of a closed point of generic fiber
Let $S= \operatorname{Spec} A$ be a local Dedekind scheme of dimension $1$, (eg spectrum of localization at a prime of the ring of integers of a number field). Let $s \in S$ it's unique closed point ...
- 5,946
1
vote
0
answers
105
views
How would you call a variety that is locally a complete intersection up to defect c?
Let $X$ be an equidimensional variety of dimension $n$ over a field that can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ (for a large enough $N$; we ...
- 16.9k
1
vote
1
answer
603
views
Morphism between Fourier-Mukai functors implies the morphism between kernels?
Suppose $X,Y$ are smooth varieties over $\mathbb{C}$, and let $K_i \in D^b(X \times Y), i=1,2$ be objects in the derived category of bounded complex of coherent sheaves on $X \times Y$. Then there are ...
- 3,472