A Noether-Lefschetz theorem for varieties of \(r\)-planes in complete intersections (Q2894134)

Let \(X\) be a complete intersection of type \((d_1, \dots, d_s)\) in the \(n\)-dimensional complex projective space, and \(r\) be a positive integer number. \(F_r(X)\) denotes the Fano scheme of \(r\)-dimensional linear spaces contained in \(X\). The expected dimension of \(F_r(X)\) is \(\delta=(r+1)(n-r)-\Sigma_i{{d_i+r}\choose{r}}\). Set \(\delta_-=\text{min}\{\delta, n-2r-s\}\). The following result is due to \textit{C. Borcea} [Pac. J. Math. 143, No. 1, 25--36 (1990; Zbl 0731.14029)], \textit{L. Bonavero} and \textit{C. Voisin} [C. R. Acad. Sci., Paris, Sér. I, Math. 323, No. 9, 1019--1024 (1996; Zbl 0951.14027)] and \textit{O. Debarre} and \textit{L. Manivel} [Math. Ann. 312, No. 3, 549--574 (1998; Zbl 0913.14015)]: if \(X\) is a general complete intersection of type \((d_1, \dots , d_s)\) with \(d_i\geq 2\) for all \(i\), then if \(\delta_-\geq 1\) \(F_r(X)\) is connected and smooth of dimension \(\delta\). If \(\delta_-\geq 3\), the Picard number of the Fano scheme is \(\rho(F_r(X))=1\).NEWLINENEWLINEThe main theorem of the article under review completes the picture considering the remaining cases. The precise result says that, if \(X\) is a very general complete intersection as above with \(\delta\geq 2\) and \(\delta_-\geq 1\), then \(\rho(F_r(X))=1\) unless \(X\) is one of the following:NEWLINENEWLINE(i) a quadric in \(\mathbb P^{2r+3}\), \(r\geq 1\), in which case \(\rho(F_r(X))=2\);NEWLINENEWLINE(ii) the complete intersection of two quadrics in \(\mathbb P^{2r+4}\), \(r\geq 1\), in which case \(\rho(F_r(X))=2r+6\).NEWLINENEWLINEA discussion of some of the exceptions is given. The proof of the theorem uses a deformation argument, which reduces it to the computation of some cohomology groups. Then one can use the Theorem of Bott on the cohomology groups of the homogeneous bundles on Grassmannians.NEWLINENEWLINEAs an application the author proves the following: let \(Z\) be a general cubic fivefold, and \(\alpha: F_2(Z)\rightarrow JZ\) be its Abel-Jacobi map. Then \([\alpha_*(F_2(Z))] =12[{\frac{\Theta^{19}}{19!}}]\).