Bigraded Hilbert functions of 0-dimensional schemes in P1xP1

Seminario congiunto Dipartimento di Matematica - Centro Internazionale per la Ricerca Matematica-Fbk

Giovedì 12 luglio 2018
Versione stampabile

Luogo: Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Sala Seminari 1

Ore: 12:00

Relatore:

  • Alessandro Oneto (Universitat Politècnica de Catalunya, and Barcelona Graduate School of Mathematics)

Abstract:

Polynomial interpolation problems have been largely studied in algebraic geometry and commutative algebra. The classical question asks what
is the dimension of a linear system of hypersurfaces in P^n of given degree and with multiple base points. The case of general double points has a very long history, which goes back to the classical school of algebraic geometry of the XIX century. However, it has been completely solved by J. Alexander and A. Hirschowitz only in 1995, by using the so-called méthode d'Horace différentiel. For higher multiplicities of the base points, even the case of planar curves is in general open. A conjectural answer to the latter case is given by the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture.

We consider a multi-graded version of the problem: what is the dimension of a linear system of curves in P1xP1 of given bidegree and with multiple base points? In 2005, M.V. Catalisano, A.V. Geramita and A. Gimigliano introduced themultiprojective-affine-projective method that reduces this problem to the classical case of fat points in P2 and solved the case of general double points.
After a general introduction, in this talk, I will present the solution to the case of triple points, and some partial result to higher multiplicities, by using a combination of the multiprojective-affine-projective method and the méthode d'Horace différentiel. If time permits, I will also explain how to use these ideas to study the Hilbert function of other types of 0-dimensional schemes and to prove that the tangential varieties to any Segre-Veronese embedding of P1xP1 are never defective.

These are joint works with E. Carlini and M. V. Catalisano.

Referente: Alessandra Bernardi