On submaximal plane curves

We prove that a submaximal curve in P has sequence of multiplicities (μ, ν, . . . , ν), with μ < sν for every integer s with (s− 1)(s+ 2) ≥ 6.76( r − 1). This note is a sequel to [10], where a specialization method was developed in order to bound the degree of singular plane curves. The problem under consideration is, given a system of multiplicities (m) = (m1,m2, . . . ,mr) ∈ Z and points p1, . . . , pr ∈ P, which we shall always assume to be in general position, to determine the minimal degree α(m) of a curve with multiplicity mi at each point pi. In [10], the focus was on homogeneous (m), (i.e., m1 = m2 = · · · = mr), but the method applies in general; here it is used to show that if one of the multiplicities is much bigger than the others, in a sense we make precise below (see theorem 1), then α(m) > ∑r i=1 mi √ r . (1) In connection with his solution to the fourteenth problem of Hilbert, M. Nagata conjectured in 1959 that the inequality (1) holds for all (m) provided r > 9, and proved it in the case when r is a perfect square (see [7] or [8]). Since then, many partial results have been proved by several authors (see for instance [13], [3], [4], [5], [11], [12]), but as far as we know the conjecture remains open in general. One of the research lines in this area is the study of submaximal curves that arise in the context of Seshadri constants. A submaximal curve is an irreducible curve which causes the (r-point) Seshadri constant of a surface to be non-maximal; in the case of P it is just an irreducible curve which causes (1) to fail, and Nagata’s conjecture can be equivalently stated by saying that there exist no submaximal curves for r > 9. T. Szemberg proved in [11, 4.6] that every submaximal curve on a surface with Picard number ρ = 1 whose multiple points are in general position is quasi-homogeneous, i.e., has m2 = · · · = mr for a suitable ordering of the points. In the case of the projective plane, our result shows that moreover m1 can not be much bigger than m2, constraining further the range of possible counterexamples to Nagata’s conjecture. It is worth mentioning that quasihomogeneous curves are relevant also for the method of C. Ciliberto and R. Miranda [2] to compute the dimension of (homogeneous) linear systems. The approach is based on the specialization introduced in [10]. Roughly speaking, one proves that if there exists a curve with given multiplicities at r general points, then