83 research outputs found
Discrepancies of non-\Q-Gorenstein varieties
We give an example of a non \Q-Gorenstein variety which is canonical but
not klt, and whose canonical divisor has an irrational valuation. We also give
an example of an irrational jumping number and we prove that there are no
accumulation points for the jumping numbers of normal non-\Q-Gorenstein
varieties with isolated singularities.Comment: 12 pages - minor change
A note on the birational geometry of tropical line bundles
Given a closed subvariety Y of a n-dimensional torus, we study how the
tropical line bundles of Trop(Y) can be induced by line bundles living on a
tropical compactification of Y in a toric variety, following the construction
of Jenia Tevelev. We then consider the general structure with respect to the
Zariski--Riemann space.Comment: Version
Ample Weil divisors
We define and study positivity (nefness, amplitude, bigness and
pseudo-effectiveness) for Weil divisors on normal projective varieties. We
prove various characterizations, vanishing and non-vanishing theorems for
cohomology, global generation statements, and a result related to log Fano.Comment: Version 3: published versio
Doctor of Philosophy
dissertationWe give an example of a non Q-Gorenstein variety whose canonical divisor has an irrational valuation and an example of a non Q-Gorenstein variety which is canonical but not klt. We also give an example of an irrational jumping number and we prove that there are no accumulation points for the jumping numbers of normal non-Q-Gorenstein varieties with isolated singularities. We prove that the canonical ring of a canonical variety in the sense of [dFH09] is finitely generated. We prove that canonical varieties are klt if and only if R(−KX) is finitely gener-ated. We introduce a notion of nefness for non-Q-Gorenstein varieties and study some of its properties. We then focus on the properties of non-Q-Gorenstein toric varieties, with particular attention to minimal log discrepancies
On positivity and base loci of vector bundles
The aim of this note is to shed some light on the relationships among some
notions of positivity for vector bundles that arose in recent decades.
Our purpose is to study several of the positivity notions studied for vector
bundles with some notions of asymptotic base loci that can be defined on the
variety itself, rather than on the projectivization of the given vector bundle.
We relate some of the different notions conjectured to be equivalent with the
help of these base loci, and we show that these can help simplify the various
relationships between the positivity properties present in the literature.
In particular, we define augmented and restricted base loci and of a vector bundle on the variety , as
generalizations of the corresponding notions studied extensively for line
bundles. As it turns out, the asymptotic base loci defined here behave well
with respect to the natural map induced by the projectivization of the vector
bundle
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