Special prime Fano fourfolds of degree 10 and index 2. (English) Zbl 1326.14094
Hacon, Christopher D. (ed.) et al., Recent advances in algebraic geometry. A volume in honor of Rob Lazarsfeld’s 60th birthday. Based on the conference, Ann Arbor, MI, USA, May 16–19, 2013. Cambridge: Cambridge University Press (ISBN 978-1-107-64755-8/pbk; 978-1-107-41600-0/ebook). London Mathematical Society Lecture Note Series 417, 123-155 (2014).
Summary: We analyze (complex) prime Fano fourfolds of degree 10 and index 2. S. Mukai [Algebraic geometry and commutative algebra, in Honor of Masayoshi Nagata, Vol. I, 357–377 (1988; Zbl 0701.14044)] gave a complete geometric description; in particular, most of them are contained in a Grassmannian \(\mathrm{Gr}(2,5)\). As in the case of cubic fourfolds, they are unirational and some are rational, as already remarked by L. Roth [Ann. Mat. Pura Appl., IV. Ser. 29, 91–97 (1949; Zbl 0036.22502)].
We show that their middle cohomology is of \(K3\) type and that their period map is dominant, with smooth four-dimensional fibers, onto a twenty-dimensional bounded symmetric period domain of type IV. Following B. Hassett [Compos. Math. 120, No. 1, 1–23 (2000; Zbl 0956.14031)], we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian \(G(2,5)\). Special fourfolds correspond to a countable union of hypersurfaces (the Noether-Lefschetz locus) in the period domain, labelled by a positive integer \(d\). We describe special fourfolds for some low values of \(d\). We also characterize those integers \(d\) for which special fourfolds do exist.
For the entire collection see [Zbl 1318.14002].
We show that their middle cohomology is of \(K3\) type and that their period map is dominant, with smooth four-dimensional fibers, onto a twenty-dimensional bounded symmetric period domain of type IV. Following B. Hassett [Compos. Math. 120, No. 1, 1–23 (2000; Zbl 0956.14031)], we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian \(G(2,5)\). Special fourfolds correspond to a countable union of hypersurfaces (the Noether-Lefschetz locus) in the period domain, labelled by a positive integer \(d\). We describe special fourfolds for some low values of \(d\). We also characterize those integers \(d\) for which special fourfolds do exist.
For the entire collection see [Zbl 1318.14002].
MSC:
| 14J45 | Fano varieties |
