We study supersolvable line arrangements in P 2 over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil) complex line arrangement cannot have more than 4 modular points, and if all of the crossing points of a complex line arrangement have multiplicity 3 or 4, then the arrangement must have 0 modular points (i.e., it cannot be supersolvable). This provides at least a little evidence for our conjecture that every nontrivial complex supersolvable line arrangement has at least one point of multiplicity 2, which in turn is a step toward the much stronger conjecture of Anzis and Tohǎneanu that every nontrivial complex supersolvable line arrangement with s lines has at least s/2 points of multiplicity 2.

AbstractLet X be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of X. Following the ideas of [T. Bauer and M. Funke, Weyl and Zariski chambers on K3 surfaces, Forum Math. 24 2012, 3, 609–625] and [S. A. Rams and T. Szemberg, When are Zariski chambers numerically determined?, Forum Math. 28 2016, 6, 1159–1166], we study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.

We prove new results on single point Seshadri constants for ample line bundles on hyperelliptic surfaces, motivated by the results in [10]. Given a hyperelliptic surface X and an ample line bundle L on X, we show that the least Seshadri constant ε(L) of L is a rational number when X is not of type 6. We also prove new lower bounds for the Seshadri constant ε(L, 1) of L at a very general point.

a b s t r a c tThe toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of locally toroidal morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface.

Let X be the blow up of P 2 at r general points p 1 , . . . , p r ∈ P 2 . We study line bundles on X given by plane curves of degree d passing through p i with multiplicity at least m i . Motivated by results in [ST3], we establish conditions for ampleness, very ampleness and global generation of such line bundles.