On Fano schemes of linear spaces of general complete intersections
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Archiv der Mathematik
On Fano schemes of linear spaces of general complete intersections Francesco Bastianelli, Ciro Ciliberto, Flaminio Flamini , and Paola Supino Abstract. We consider the Fano scheme Fk (X) of k-dimensional linear subspaces contained in a complete intersection X ⊂ Pn of multi-degree d = (d1 , . . . , ds ). Our main result is an extension of a result of Riedl and Yang concerning Fano schemes of lines on very general hypersurfaces: we consider the case when X is a very general complete intersection and Πsi=1 di > 2 and we find conditions on n, d, and k under which Fk (X) does not contain either rational or elliptic curves. At the end of the paper, we study the case Πsi=1 di = 2. Mathematics Subject Classification. Primary 14M10; Secondary 14C05, 14M15, 14M20. Keywords. Complete intersections, Parameter spaces, Fano schemes, Rational and elliptic curves.
1. Introduction. In this paper, we are concerned with the Fano scheme Fk (X) ⊂ G(k, n), parameterizing k-dimensional linear subspaces contained in X ⊂ Pn , when X is a complete intersection of multi-degree d = (d1 , . . . , ds ), with 1 s n − 2. We will avoid the trivial case in which X is a linear subspace, so that Πsi=1 di 2. Our inspiration has been the following result by Riedl and Yang concerning the case of hypersurfaces: Theorem 1.1 (cf. [8, Thm. 3.3]). If X ⊂ Pn is a very general hypersurface of , then F1 (X) contains no rational curves. degree d such that n (d+1)(d+2) 6 This collaboration has benefitted of funding from the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP: E83C18000100006).
Arch. Math.
F. Bastianelli et al.
This paper is devoted to generalize Riedl–Yang’s result to complete intersections and to arbitrary k1: Theorem 1.2 Let X ⊂ Pn be a very general complete intersection of multidegree d = (d1 , . . . , ds ), with 1 s n − 2 and Πsi=1 di > 2. Let 1 k n − s − 1 be an integer. If s 1 di + k + 1 n k−1+ , (1.1) k + 2 i=1 k+1 then Fk (X) contains neither rational nor elliptic curves. The proof is contained in Section 3. Section 4 concerns the quadric case Πsi=1 di = 2. We work over the complex field C. As customary, the term “general” is used to denote a point which sits outside a union of finitely many proper closed subsets of an irreducible algebraic variety whereas the term “very general” is used to denote a point sitting outside a countable union of proper closed subsets of an irreducible algebraic variety. , . . . , ds ) be an s-tuple 2. Preliminaries. Let n 3, 1 s n − 2, and d = (d1 s of positive integers such that Πsi=1 di 2. Let Sd := i=1 H 0 (Pn , OPn(di )) s and consider its Zariski open subset Sd∗ := i=1 H 0 (Pn , OPn (di )) \ {0} . For any u := (g1 , . . . , gs ) ∈ Sd∗ , let Xu := V (g1 , . . . , gs ) ⊂ Pn denote the closed subscheme defined by the vanishing of the polynomials g1 , . . . , gs . When u ∈ Sd∗ is general, Xu is a smooth, irreducible variety of dimension n − s 2, so that Sd∗ contains an open dense subse
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