On complete intersections containing a linear subspace
| Autor: | Bastianelli, Francesco, Ciliberto, Ciro, Flamini, Flaminio, Supino, Paola |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | Consider the Fano scheme $F_k(Y)$ parameterizing $k$-dimensional linear subspaces contained in a complete intersection $Y \subset \mathbb{P}^m$ of multi-degree $\underline{d} = (d_1, \ldots, d_s)$. It is known that, if $t := \sum_{i=1}^s \binom{d_i +k}{k}-(k+1) (m-k)\leqslant 0$ and $\Pi_{i=1}^sd_i >2$, for $Y$ a general complete intersection as above, then $F_k(Y)$ has dimension $-t$. In this paper we consider the case $t> 0$. Then the locus $W_{\underline{d},k}$ of all complete intersections as above containing a $k$-dimensional linear subspace is irreducible and turns out to have codimension $t$ in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general $[Y]\in W_{\underline{d},k}$ the scheme $F_k(Y)$ is zero-dimensional of length one. This implies that $W_{\underline{d},k}$ is rational. Comment: 6 pages, the collaboration has benefitted of funding from the research project \emph{"Families of curves: their moduli and their related varieties"} (CUP: E81-18000100005) - Mission Sustainability - University of Rome Tor Vergata |
| Databáze: | arXiv |
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