13 research outputs found
A divisorial valuation with irrational volume
In this paper we present a divisorial valuation with irrational volume using
an algebro-geometric construction.Comment: 8 pages, 2 figure
Asymptotic cohomological functions on projective varieties
In this paper we define certain analogues of the volume of a divisor - called
asymptotic cohomological functions - and investigate their behaviour on the
Neron--Severi space. We establish that asymptotic cohomological functions are
invariant with respect to the numerical equivalence of divisors, and that they
give rise to continuous functions on the real Neron--Severi space. To
illustrate the theory, we work out these invariants for abelian varieties,
smooth surfaces, and certain homogeneous spaces.Comment: 32 pages, 3 figure
Volume functions of linear series
The volume of a Cartier divisor is an asymptotic invariant, which measures
the rate of growth of sections of powers of the divisor. It extends to a
continuous, homogeneous, and log-concave function on the whole N\'eron--Severi
space, thus giving rise to a basic invariant of the underlying projective
variety. Analogously, one can also define the volume function of a possibly
non-complete multigraded linear series. In this paper we will address the
question of characterizing the class of functions arising on the one hand as
volume functions of multigraded linear series and on the other hand as volume
functions of projective varieties. In the multigraded setting, relying on the
work of Lazarsfeld and Musta\c{t}\u{a} (2009) on Okounkov bodies, we show that
any continuous, homogeneous, and log-concave function appears as the volume
function of a multigraded linear series. By contrast we show that there exists
countably many functions which arise as the volume functions of projective
varieties. We end the paper with an example, where the volume function of a
projective variety is given by a transcendental formula, emphasizing the
complicated nature of the volume in the classical case.Comment: 16 pages, minor revisio
Asymptotic cohomological functions of toric divisors
We study functions on the class group of a toric variety measuring the rates
of growth of the cohomology groups of multiples of divisors. We show that these
functions are piecewise polynomial with respect to finite polyhedral chamber
decompositions. As applications, we express the self-intersection number of a
T-Cartier divisor as a linear combination of the volumes of the bounded regions
in the corresponding hyperplane arrangement and prove an asymptotic converse to
Serre vanishing.Comment: 13 pages. v2: corrected typos, minor revisions. To appear in Adv.
Mat