Impact
~29k
people reached
 0 posts edited
 0 helpful flags
 84 votes cast
12h

answered  Existense of a semistable vector bundle on smooth curves in positive characteristic 
1d

comment 
Which sheaves on a projective bundle are flat over the base scheme?
Also, if you are just interested in sheaves such that $f^*f_*G\to G$ is an isomorphism, you can conclude that there are no higher direct images without needing flatness (in this situation). Namely, such sheaves are exactly sheaves of the form $f^*H$ for some $H$ (not necessarily locally free), and use the projection formula. 
Oct
2 
comment 
Quantum cohomology of line bundles over $\mathbb P^N$
Do you mean the action of $GL(N)$? ($O(n)$ is generally speaking not equivariant for $PGL(N)$) 
Oct
1 
reviewed  Reviewed A Question on Chinese Remainder Theorem 
Oct
1 
awarded  Custodian 
Oct
1 
reviewed  Approve Theorem of Bryant in higher dimensions 
Oct
1 
comment 
A Nodal curve embedded in a smooth variety, is always regularly embedded?
I think so, it is a local complete intersection, like Allen Knutson says. 
Oct
1 
comment 
A Nodal curve embedded in a smooth variety, is always regularly embedded?
No, it's not possible: look at the generators of the ideal of C, and choose among those the elements whose differentials at p are linearly independent. They will cut out a subvariety that is smooth at p, and its tangent space is the same as C, which makes it a surface. 
Oct
1 
awarded  Custodian 
Oct
1 
reviewed  Reviewed Embedding of parallelizable closed smooth manifold 
Sep
30 
comment 
$(L, \nabla)$ comes from a $G$bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?
By the Tannakian formalism, a Gconnection on A is the same as a morphism H→G. Since H is commutative, the morphism factors through a commutative subgroup of G. 
Sep
30 
comment 
$(L, \nabla)$ comes from a $G$bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?
(Sorry, I keep misreading things and having to delete comments. Hopefully this makes sense:) Consider the category of vector bundles with connections on A. It is Tannakian with the fiber functor, say, fiber at 0. By the Fourier transform, it is equivalent to the category of torsion sheaves on A♭. We now see that it is actually a category of representations of a commutative proalgebraic group H (which explicitly looks like something horrible: the Cartier dual of the group which is a disjoint union of completions of A♭ at all points)... 
Sep
30 
awarded  ag.algebraicgeometry 
Sep
29 
comment 
$(L, \nabla)$ comes from a $G$bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?
The question was for $GL(n)$ :) It also seems that once you prove it for $GL(n)$, it extends to other groups automatically (basically by Tanakian formalism). Maybe there is a shorter way? 
Sep
29 
answered  $(L, \nabla)$ comes from a $G$bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$? 
Sep
22 
comment 
vanishing locus of a certain section of $\wedge^{2n} (\mathcal{O}^{2n}_{M \times M})$
If M is higherdimensional, the vanishing locus cannot be diagonal for dimensional reasons: the vanishing locus is a divisor, while the diagonal has codimension = dimension of M. 
Sep
22 
comment 
vanishing locus of a certain section of $\wedge^{2n} (\mathcal{O}^{2n}_{M \times M})$
Actually, thinking about it a bit, it seems that if the curve is not rational, there will always be nondiagonal zeros (simply from looking at the divisor of the determinant). So, for (1B) for projective curve, the story seems to be:  If M is nonrational, there will always be pairs of distinct points where the two vector spaces meet;  If M is rational, there is a way to choose the family of vector spaces so that the zero locus of $y$ is (nth multiple of) the diagonal, but it is certainly not going to hold for all nonconstant families. 
Sep
22 
comment 
vanishing locus of a certain section of $\wedge^{2n} (\mathcal{O}^{2n}_{M \times M})$
The way the question is currently formulated, you seem to be asking about (2A') However, it is easy to find counterexamples to version (A,A'), and so far I don't see what condition in (A") would create a nontrivial question. To me at least, the most interesting would be (1B), which is basically a question about curves in Grassmannian. One advantage of (1) over (2) is that it makes sense for projective curves, which (at least to me) are more interesting. If the curve is rational, the answer to (1B) is clearly `yes', in general, it seems to be an interesting question. So: what is it you want? 
Sep
22 
comment 
vanishing locus of a certain section of $\wedge^{2n} (\mathcal{O}^{2n}_{M \times M})$
Frankly, it is still pretty hard to figure out exactly what it is you are asking... but there may be some interesting questions in this direction. Let me try: Let V be a subbundle of the trivial bundle $O^{2n}_M$ (version 1: any subbundle, version 2:subbundle with a global basis). Consider the condition that $V_x$ and $V_y$ are transversal whenever $x\ne y$. Does this condition hold (version A: for all V; version A': for all nonconstant V; version A": for all V satisfying some as yet unclear extra condition; version B: for some V). (to be continued) 
Sep
22 
comment 
Ramification of the map from the stack of elliptic curves to the $j$line
One way is to notice that at these points, the automorphisms act nontrivially on the tangent space to the stack. This implies that the differential of j is zero at these points, and therefore j is not etale. 