On the Terracini Locus of Projective Varieties

We introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite sets $$S \subset X$$ S ⊂ X such that 2S fails to impose independent conditions to a linear system L. Terracini loci are relevant in the study of interpolation problems over double points in special position, but they also enter naturally in the study of special loci contained in secant varieties to projective varieties.We find some criteria which exclude that a set S belongs to the Terracini locus. Furthermore, in the case where X is a Veronese variety, we bound the dimension of the Terracini locus and we determine examples in which the locus has codimension 1 in the symmetric product of X.

On blowup of secant varieties of curves

<p style='text-indent:20px;'>In this paper, we show that for a nonsingular projective curve and a positive integer <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula>, the <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-th secant bundle is the blowup of the <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>-th secant variety along the <inline-formula><tex-math id="M4">\begin{document}$ (k-1) $\end{document}</tex-math></inline-formula>-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.</p>

A Characterization of Secant Varieties of Severi Varieties Among Cubic Hypersurfaces

It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.