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

[摘要] Denoting by L-d(m(0), m(1), . . . ., m(r)) the linear system of plane curves of degree d passing through r + 1 generic points p(0). p(1)..... p(r) of the projective plane with multiplicity m(i) (or larger) at each pi, we prove the Harbourne-Hirschowitz Conjecture for linear systems L-d(m(0), m(1), . . . . m(r)) determined by a wide family of systems of multiplicities m = (m(i))(i)(r)= 0 and arbitrary deeree d. Moreover, we provide an algorithm for computing a bound for the regularity of an arbitrary system m, and we give its exact value when 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. (c) 2007 Elsevier B.V. All rights reserved.

[发布日期] 2007-12-01