Variations on a conjecture of Nagata
 
 
Description:  The classical multivariate Hermite interpolation problem asks for the determination of the dimension of the vector space of polynomials of a given degree d in a given number r of variables having n assigned, sufficiently general, zeroes with given multiplicities. This is trivial for n=1, but quite complicated as soon as n\ge 2. Very little is known for n>2. For n=2 there is a leading conjecture going back to B. Segre in 1960 (the so called Segre-Harbourne-Gimigliano-Hirschowitz conjecture). In the same year Nagata, working on Hilbert's XIV problem for which he provided a counterexample, formulated a related conjecture, which turns out to be implied by SHGH, and, in a sense which can be made precise, ''asymptotically equivalent'' to it. In this talk I will recall these conjectures, and I will try to explain their connections with delicate properties of sophisticated objects like the Mori cone of the blow-up of a plane at ten or more general points. This viewpoint suggests a natural intermediate conjecture, i.e it is implied by SHGH and implies Nagata. I will give some little evidence for it with applications to Nagata's conjecture and to the construction of more counterexamples to Hilbert's XIV problem.
The final part of the talk is joint work with B. Harbourne, R. Miranda and J. Roé.
Start Date:  2012-03-28
Start Time:   15:00
Speaker:  Ciro Ciliberto (Univ. degli Studi di Roma Tor Vergata, Italy)
Institution:  Univ. degli Studi di Roma Tor Vergata
Place:  Sala 2.4
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