Linear subspaces of hypersurfaces (Q821480)

Let \(X\) be a smooth hypersurface of degree \(d\) in \(\mathbb{CP}^n\). The Hilbert scheme parametrizing \(k\) dimensional linear subspaces contained in \(X\) is denoted \(F_k(X)\). In particular, \(F_1(X)\) parametrizes lines contained in \(X\) (commonly know as Fano variety of lines on \(X\)). A well known conjecture in the field, due to Debarre and de Jong, states that the variety of lines \(F_1(X)\) has the expected dimension \(2n-3-d\) when \(n\geq d\). It is a classically known result in the field that the conjecture holds for \(d=3\). \textit{A. Collino} [J. Lond. Math. Soc., II. Ser. 19, 257--267 (1979; Zbl 0432.14024)] proved the conjecture for \(d=4\). In an unpublished work, Debarre showed the conjecture to be true when \(d\leq 5\). In [Duke Math. J. 95, No. 1, 125--160 (1998; Zbl 0991.14018)] the three authors proved the statement when \(d\) is small with respect to \(n\). The cases \(d\leq 6\) was settled in [\textit{R. Beheshti}, J. Reine Angew. Math. 592, 1--21 (2006; Zbl 1094.14029); [\textit{J. M. Landsberg} and \textit{C. Robles}, J. Lond. Math. Soc., II. Ser. 82, No. 3, 733--746 (2010; Zbl 1221.14058); \textit{J. Landsberg} and \textit{O. Tommasi}, Mich. Math. J. 59, No. 3, 573--588 (2010; Zbl 1209.14035)]. The cases \(d\leq 8\) were proven by \textit{R. Beheshti} [Math. Ann. 360, No. 3--4, 753--768 (2014; Zbl 1304.14065)]. In the paper under review, the two authors prove the Debarre-de Jong conjecture when \(n\geq 2d-4\). The key ingredient is a result (Lemma 2.1) that provides an upper bound to the dimension of the locus of tangency of a smooth hypersurface with varieties cut out by lower degree equations. The authors also prove a similar result for the space \(F_k(X)\). Namely, they show that if \(n\geq 2\binom{d+k-1}{k}+k\) then \(F_k(X)\) is irreducible of the expected dimension. As an application, it is proven the unirationality of smooth hypersurfaces for which the inequality \(n\geq 2^{d!}\) holds.