arXiv subjects
Publications and source records attributed to Joaquim Roé.
In this paper we give a new proof of the fact that for all pairs of positive integers (d, m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty.
We prove that general unions of singularity schemes of multiplicity two in the projective plane have maximal rank.
T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagata's 1959 open conjecture, which claims that the Seshadri constant of r>9 very general points of the projective plane is maximal. Here we prove that Nagata's original conjecture implies Szemberg's for all smooth surfaces X with an ample divisor L generating its Neron-Severi group and such that L^2 is a square. More generally, we prove that the (n-1)-dimensional Seshadri constant of an ample divisor L on a projective variety X of dimension n at r very general points is bounded below by the product of the (n-1)-dimensional Seshadri constant of at a very general point times the (n-1)-dimensional Seshadri constant of the hyperplane on projective n-space at r very general points.
We prove that a submaximal plane curve (i.e., an irreducible counterexample to Nagata's conjecture) with r singular points has sequence of multiplicities (m, n, ..., n) with m 6.76(r-1).
Given positive integers $m_1, m_2, ..., m_n$, and $n$ general points $p_i$ of ${\bf CP}^2$, bounds are given for the least degree $t$ among plane curves passing through each point $p_i$ with multiplicity at least $m_i$, and for the least $t$ such that the $n$ multiple points impose independent conditions on curves of degree $t$, often improving substantially what was previously known. As an application, the Hilbert function (resp., minimal free resolution) is determined for symbolic powers $I^{(m)}$ for the ideal $I$ defining $n$ general points of ${\bf CP}^2$ for infinitely many m for each square n (resp., for infinitely many m for each even square n). Four graphs are included showing other values of m and n for which results are given.
Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points of P2 are verified for infinitely many m for each square n > 9, using an approach developed by the authors in a previous paper. In those cases that n is even, conjectures for the resolution are also verified. Previously, by work of Evain (for the Hilbert function) and of Harbourne, Holay and Fitchett (for the resolution), these conjectures were known for infinitely many m for a given n > 9 only for n being a power of 4.