Curves having one place at infinity and linear systems on rational surfaces

Denoting by ${\mathcal L}_d(m_0,m_1,...,m_r)$ the linear system of plane curves passing through $r+1$ generic points $p_0,p_1,...,p_r$ of the projective plane with multiplicity $m_i$ (or larger) at each $p_i$, we prove the Harbourne-Hirschowitz Conjecture for linear systems ${\mathcal L}_d(m_0,m_1,...,m_r)$ determined by a wide family of systems of multiplicities $\bold{m}=(m_i)_{i=0}^r$ and arbitrary degree $d$. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system $\bold{m}$ and we give its exact value when $\bold{m}$ is in the above family. To do that, we prove an $H^1$-vanishing theorem for line bundles on surfaces associated with some pencils ``at infinity''.Comment: This is a revised version of a preprint of 200

Toric bases for 6D F-theory models

We find all smooth toric bases that support elliptically fibered Calabi-Yau threefolds, using the intersection structure of the irreducible effective divisors on the base. These bases can be used for F-theory constructions of six-dimensional quantum supergravity theories. There are 61,539 distinct possible toric bases. The associated 6D supergravity theories have a number of tensor multiplets ranging from 0 to 193. For each base an explicit Weierstrass parameterization can be determined in terms of the toric data. The toric counting of parameters matches with the gravitational anomaly constraint on massless fields. For bases associated with theories having a large number of tensor multiplets, there is a large non-Higgsable gauge group containing multiple irreducible gauge group factors, particularly those having algebras e_8, f_4 and (g_2 + su(2)) with minimal (non-Higgsable) matter.Comment: 39 pages, 13 figures, one appendix; ancillary data file contains list of 61,539 bases; v2: minor correctio

On a notion of speciality of linear systems in P^n

Given a linear system in P^n with assigned multiple general points we compute the cohomology groups of its strict transforms via the blow-up of its linear base locus. This leads us to give a new definition of expected dimension of a linear system, which takes into account the contribution of the linear base locus, and thus to introduce the notion of linear speciality. We investigate such a notion giving sufficient conditions for a linear system to be linearly non-special for arbitrary number of points, and necessary conditions for small numbers of points.Comment: 26 pages. Minor changes, Definition 3.2 slightly extended. Accepted for publication in Transactions of AM

Newton polygons and curve gonalities

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special cases. One proof technique uses recent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture to a purely combinatorial statement.Comment: 29 pages, 18 figures; erratum at the end of the articl