69 research outputs found
Rational families of vector bundles on curves, I
Let C be a smooth complex projective curve of genus at least 2 and let M be
the moduli space of rank 2, stable vector bundles on C, with fixed determinant
of degree 1. For any k>1, we find two irreducible components of the space of
rational curves of degree k on M. One component, which we call the nice
component has the property that the general element is a very free curve if k
is sufficiently large. The other component has the general element a free
curve. Both components have the expected dimension and their maximal rationally
connected fibration is the Jacobian of the curve C.Comment: 23 page
Polarized minimal families of rational curves and higher Fano manifolds
In this paper we investigate Fano manifolds whose Chern characters
satisfy some positivity conditions. Our approach is via the study of
polarized minimal families of rational curves through a general
point . First we translate positivity properties of the Chern
characters of into properties of the pair . This allows us to
classify polarized minimal families of rational curves associated to Fano
manifolds satisfying and . As a first
application, we provide sufficient conditions for these manifolds to be covered
by subvarieties isomorphic to and . Moreover, this
classification enables us to find new examples of Fano manifolds satisfying
.Comment: 17 page
Exceptional collections on certain Hassett spaces
We construct an invariant full exceptional collection on
Hassett spaces of weighted stable rational curves with markings and
weights , for
and can be identified with symmetric GIT quotients of
by the diagonal action of when is odd,
and their Kirwan desingularization when is even. The existence of such an
exceptional collection is one of the needed ingredients in order to prove the
existence of a full -invariant exceptional collection on
. To prove exceptionality we use the method of
windows in derived categories. To prove fullness we use previous work on the
existence of invariant full exceptional collections on Losev-Manin spaces.Comment: At the request of the referee, the paper arXiv:1708.06340 has been
split into two parts. This is the second of those papers (submitted). 36
page
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