6533b86ffe1ef96bd12cd106

RESEARCH PRODUCT

Real Line Arrangements and Surfaces with Many Real Nodes

subject

description

A long standing question is if the maximum number μ(d) of nodes on a surface of degree d in P( ) can be achieved by a surface defined over the reals which has only real singularities. The currently best known asymptotic lower bound, μ(d) 5 12 d, is provided by Chmutov’s construction from 1992 which gives surfaces whose nodes have non-real coordinates. Using explicit constructions of certain real line arrangements we show that Chmutov’s construction can be adapted to give only real singularities. All currently best known constructions which exceed Chmutov’s lower bound (i.e., for d = 3, 4, . . . , 8, 10, 12) can also be realized with only real singularities. Thus, our result shows that, up to now, all known lower bounds for μ(d) can be attained with only real singularities. We conclude with an application of the theory of real line arrangements which shows that our arrangements are aymptotically the best possible ones for the purpose of constructing surfaces with many nodes. This proves a special case of a conjecture of Chmutov.