Projectively normal but superabundant embeddings of rational surfaces in projective space
Among the rational surfaces, those obtained by blowing up a collection of distinct points in \(\mathbb{P}^2\) have classically received the most attention. On the one hand, that attention is often focused on embeddings of the blown up surface in some projective space. On the other hand, there is much interest in the dimension of linear systems of plane curves defined by base point conditions. Such investigations usually involve blowing up the points and studying the dimension of associated complete linear systems on the blown up surface. Conjecturally, any very ample divisor on a surface obtained by blowing up sufficiently general points should be non-superabundant, which would mean that the original linear system in the plane has the dimension one expect assuming the base point conditions that define it are independent. In this paper we bring together both of these aspects of current studies of blow-ups of the plane, by identifying a well-behaved sublocus of sets of points which when blown up produce surfaces with superabundant very ample divisors, and by studying projective embeddings that result from certain of these divisors. In particular, the sets of points of interest here are complete intersections of plane curves meeting transversely. Our results combine to prove: Let \(X\) be obtained by blowing up the points of intersection of two plane curves meeting transversely. Then \(X\) supports very ample irregular divisors if and only if both curves have degree at least 4.
embeddings the blown up surface in some projective space
dimension of linear systems of plane curves
scientific article; zbMATH DE number 701276