Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture

Abstract

ABSTRACT. We study the deformation theory of lines on the Fermat quintic threefold. We formulate an infinitesimal version of the generalized Hodge conjecture, and use our analysis of lines to prove it in a special case.

Degree 1 curves in the Dwork pencil and the mirror quintic

“…Thus by counting we get 375 lines of intersection: each cone contains 15 such special lines, each special line is at the intersection of exactly 2 cones. The fact that the only lines in the Fermat quintic X 0 are those lying in one of these cones is proven in [AK1].…”

“…Note that A.Albano and S.Katz have shown in [AK1] that the local dimension of H is 2 at each point of H 0 .…”

“…We call these lines van Geemen lines, due to their use by Bert van Geemen in [AK1] to give the first proof of the fact that the relative dimension of H = Hilb 1 (X /C) is 1. It turns out that the local behavior of H at the Van Geemen lines is geometrically understandable.…”

“…1. Hilbert scheme H 0 of lines in the Fermat quintic threefold In [AK1], H 0red is proven to be the union of 50 Fermat quintic plane curves meeting transversely in pairs at 375 points. We will start by recalling the description of the Hilbert scheme H 0 as in [AK1] and [CK]:…”

“…Mirror symmetry arose in 1991, when Candelas, de la Ossa, Green and Parkes used that principle to predict invariants very closely related to the expected numbers of rational curves of any degree on a quintic threefold. In the same year, A.Albano and S. Katz gave a first description of the Hilbert scheme of lines on the Fermat quintic threefold X 0 (see [AK1]). They proved that all the lines in X 0 are actually contained in 50 cones over some plane quintics.…”