Secant Varieties of the Varieties of Reducible Hypersurfaces in ${\mathbb P}^n$
| Autor: | Catalisano, M. V., Geramita, A. V., Gimigliano, A., Harbourne, B., Migliore, J., Nagel, U., Shin, Y. S. |
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| Rok vydání: | 2015 |
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| Druh dokumentu: | Working Paper |
| Popis: | Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$ in $n$ variables, and given an integer $\ell>1$ and a partition $\lambda$ of $d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensions of the $\ell$-secant varieties $\sigma_\ell ({\mathbb X}_{n-1,\lambda})$ for the subvariety ${\mathbb X}_{n-1,\lambda} \subset V$ of hypersurfaces whose defining forms have a factorization into forms of degrees $d_1,\ldots,d_r$. Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of $\sigma_\ell({\mathbb X}_{n-1,\lambda})$ for any choice of parameters $n,\ell$ and $\lambda$. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., $r=2$), we also relate this problem to a conjecture by Fr\"oberg on the Hilbert function of an ideal generated by general forms. Comment: 48 pages; corrected a typo in the statement of Proposition 7.2 and added short explanation. Appeared in J. Algebra |
| Databáze: | arXiv |
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