28 research outputs found
Cohomology of toric line bundles via simplicial Alexander duality
We give a rigorous mathematical proof for the validity of the toric sheaf
cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B.
Jurke, T. Rahn, and H. Roschy (arXiv:1003.5217). We actually prove not only the
original algorithm but also a speed-up version of it. Our proof is independent
from (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T.
Rahn (arXiv:1006.2392), and has several advantages such as being shorter and
cleaner and can also settle the additional conjecture on "Serre duality for
Betti numbers" which was raised but unresolved in arXiv:1006.2392.Comment: 9 pages. Theorem 1.1 and Corollary 1.2 improved; Abstract and
Introduction modified; References updated. To appear in Journal of
Mathematical Physic
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
Multigraded regularity: coarsenings and resolutions
Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for
each i and suppose that M is a finitely generated Z^r-graded S-module. In this
paper we describe how to find finite subsets of Z^r containing the multidegrees
of the minimal multigraded syzygies of M. To find such a set, we first coarsen
the grading of M so that we can view M as a Z-graded S-module. We use a
generalized notion of Castelnuovo-Mumford regularity, which was introduced by
D. Maclagan and G. Smith, to associate to M a number which we call the
regularity number of M. The minimal degrees of the multigraded minimal syzygies
are bounded in terms of this invariant.Comment: 20 pages, 1 figure; small corrections made; final version; to appear
in J. of Algebr
Exact matrix formula for the unmixed resultant in three variables
We give the first exact determinantal formula for the resultant of an unmixed
sparse system of four Laurent polynomials in three variables with arbitrary
support. This follows earlier work by the author on exact formulas for
bivariate systems and also uses the exterior algebra techniques of Eisenbud and
Schreyer. Along the way we will prove an interesting new vanishing theorem for
the sheaf cohomology of divisors on toric varieties. This will allow us to
describe some supports in four or more variables for which determinantal
formulas for the resultant exist.Comment: 24 pages, 2 figures, Cohomology vanishing theorem generalized with
new combinatorial proof. Can state some cases of exact resultant formulas in
higher dimensio