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http://hdl.handle.net/2307/602
<Title>Degenerations and applications : polynomial interpolation and secant degree</Title>
<Authors>Postinghel, Elisa</Authors>
<Issue Date>2010-04-07</Issue Date>
<Abstract>The polynomial interpolation problem in several variables and higher multiplicities is a
subject that has been widely studied, but there is only a little understanding about the
question. What is known, so far, is essentially concentrated in the Alexander-Hirschowitz
Theorem which says that a general collection of double points in Pr gives independent conditions on the linear system L of the hypersurfaces of degree d, with a well known list of
exceptions. In the ﬁrst part of this thesis we present a new proof of this theorem which consists in performing degenerations of Pr and analyzing how L degenerates. Our construction
gives hope for further extensions to greater multiplicities.
There is a long tradition within algebraic geometry that studies the dimension and the
degree of k -secant varieties. These are problems that are unsolved in general. In the second
part of the thesis, we consider any projective toric surface XP associated to a polytope
P ⊆ R2 and we perform planar toric degenerations D of XP in order to study the k -secant
varieties of XP . In particular we give a lower bound to the secant degree and to the 2-secant
degree of XP , taking into account the singularities of the conﬁguration D of non-delightful
planar toric degenerations.
1</Abstract>2010-04-06T22:00:00Z