We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

Comparing powers and symbolic powers of ideals

http://staff.math.su.se/shapiro/ProblemSolving/ottaviani.pdf

On the Alexander–Hirschowitz theorem

In this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m. A conjecture of Harbourne and Hirschowitz implies that when d > 3m, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple (-1)-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all m < 12.

/pdf/linear-systems-of-plane-curves-with-base-points-of-equal-3z07kivr7z.pdf

Linear systems of plane curves with base points of equal multiplicity

An asymptotic vanishing theorem for generic unions of multiple points

At how many points is it possible to prescribe values to a polynomial of given degree, together with derivatives up to order, say, t-1? We solve this problem in case of polynomials in two variables for t=2 and t=3 and in case of polynomials in three variables for t=2. Proofs develop in the frame of modern projective geometry.

La Methode d1Horace pour l'Interpolation à Plusieurs Variables

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La Methode d1Horace pour l'Interpolation à Plusieurs Variables