Fano Varieties and Linear Sections of Hypersurfaces

When n satisfies an inequality which is almost best possible, we prove that the k-plane sections of every smooth, degree d, complex hypersurface in Pn dominate the moduli space of degree d hypersurfaces in Pk. As a corollary we prove that, for n sufficiently large, every smooth, degree d hypersurface in Pn satisfies a version of “rational simple connectedness”. 1. Statement of results In their article [2], Harris, Mazur and Pandharipande prove that for fixed integers d and k, there exists an integer n0 = n0(d, k) such that for every n ≥ n0, every smooth degree d hypersurface X in PC has a number of good properties: (i) The hypersurface is unirational. (ii) The Fano variety of k-planes in X has the expected dimension. (iii) The k-plane sections of the hypersurface dominate the moduli space of degree d hypersurfaces in P. It is this last property which we consider. To be precise, the statement is that the following rational transformation Φ : G(k, n) 99K P//PGLk+1 is dominant. Here G(k, n) is the Grassmannian parametrizing linear Ps in P, Pd is the parameter space for degree d hypersurface in P, Pd//PGLk+1 is the moduli space of semistable degree k hypersurface in P, and Φ is the rational transformation sending a k-plane Λ to the moduli point of the hypersurface Λ ∩X ⊂ Λ (assuming Λ ∩X is a semistable degree k hypersurface in P). The bound n0(d, k) is very large, roughly a d-fold iterated exponential. Our result is the following. Theorem 1.1. Let X be a smooth degree d hypersurface in P. The map Φ is dominant if n ≥ ( d+ k − 1 k ) + k − 1. Question 1.2. For fixed d and k, what is the smallest integer n0 = n0(d, k) such that for every n ≥ n0 and every smooth, degree d hypersurface in P, the associated rational transformation Φ is dominant? Theorem 1.1 is equvialent to the inequality n0(d, k) ≤ ( d+ k − 1 k ) + k − 1.