23,421 research outputs found
On pliability of del Pezzo fibrations and Cox rings
We develop some concrete methods to build Sarkisov links, starting from Mori
fibre spaces. This is done by studying low rank Cox rings and their properties.
As part of this development, we give an algorithm to construct explicitly the
coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the
generalisation of the notion of well-formedness for weighted projective spaces
to homogeneous coordinate ring of toric varieties. As an illustration, we apply
these methods to study birational transformations of certain fibrations of del
Pezzo surfaces over , into other Mori fibre spaces, using Cox
rings and variation of geometric invariant theory. We show that the pliability
of these Mori fibre spaces is at least three and they are not rational
On the motives of moduli of chains and Higgs bundles
We take another approach to Hitchin's strategy of computing the cohomology of
moduli spaces of Higgs bundles by localization with respect to the
circle-action. Our computation is done in the dimensional completion of the
Grothendieck ring of varieties and starts by describing the classes of moduli
stacks of chains rather than their coarse moduli spaces.
As an application we show that the n-torsion of the Jacobian acts trivially
on the middle dimensional cohomology of the moduli space of twisted
SL_n-Higgs-bundles of degree coprime to n and we give an explicit formula for
the motive of the moduli space of Higgs bundles of rank 4 and odd degree. This
provides new evidence for a conjecture of Hausel and Rodr\'iguez-Villegas.
Along the way we find explicit recursion formulas for the motives of several
types of moduli spaces of stable chains.Comment: 44 page
Moduli of algebraic varieties
We develop a moduli theory of algebraic varieties and pairs of non-negative
Kodaira dimension. We define stable minimal models and construct their
projective coarse moduli spaces under certain natural conditions. This can be
applied to a wide range of moduli problems in algebraic geometry.Comment: 67 page
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