536 research outputs found
Stable maps and Quot schemes
In this paper we study the relationship between two different
compactifications of the space of vector bundle quotients of an arbitrary
vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is
a moduli space of stable maps to the relative Grassmannian.
We establish an essentially optimal upper bound on the dimension of the two
compactifications. Based on that, we prove that for an arbitrary vector bundle,
the Quot schemes of quotients of large degree are irreducible and generically
smooth. We precisely describe all the vector bundles for which the same thing
holds in the case of the moduli spaces of stable maps. We show that there are
in general no natural morphisms between the two compactifications. Finally, as
an application, we obtain new cases of a conjecture on effective base point
freeness for pluritheta linear series on moduli spaces of vector bundles.Comment: 39 pages, 1 figure; final version with a few expository changes
suggested by the refere
Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties
In this paper, we associate an invariant to an algebraic
point on an algebraic variety with an ample line bundle . The
invariant measures how well can be approximated by rational points
on , with respect to the height function associated to . We show that
this invariant is closely related to the Seshadri constant
measuring local positivity of at , and in particular that Roth's theorem
on generalizes as an inequality between these two invariants
valid for arbitrary projective varieties.Comment: 55 pages, published versio
Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one
Let X=G/B be a complete flag variety, and L' and L" two line bundles on X.
Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L),
where L=L' x L" and d=d'+d".
We answer two natural questions about the map above: When is it a nonzero map
of irreducible G-representations? Conversely, given generic irreducible
representations V' and V" of G, which irreducible components of V' x V" may
appear in the right hand side of the map above? We also give bounds on the
multiplicities appearing in a tensor product, and relate these considerations
to the boundary of the Littlewood-Richardson cone.Comment: 61 pages, 2 figures, uses PStrick
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