2011/09/07 by Collino, A., Naranjo, J. C., Pirola, G. P.
#14C25 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1109.1456
The Fano surface F of lines in the cubic threefold V is naturally embedded in the intermediate Jacobian J(V), we call "Fano cycle" the difference F-F-, this is homologous to 0 in J(V). We study the normal function on the moduli space which computes the Abel-Jacobi image of the Fano cycle. By means of the related infinitesimal invariant we can prove that the primitive part of the normal function is not of torsion. As a consequence we get that, for a general V, F-F- in not algebraically equivalent to zero in J(V) (already proved by van der Geer-Kouvidakis) and, moreover, there is no a divisor in JV containing both F and F- and such that these surfaces are homologically equivalent in the divisor. Our study of the infinitesimal variation of Hodge structure for V produces intrinsically a threefold Ξ(V) in \mathbb G the Grasmannian of lines in \mathbb P4. We show that the infinitesimal invariant at V attached to the normal function gives a section for a natural bundle on Ξ(V) and more specifically that this section vanishes exactly on Ξ∩ F, which turns out to be the curve in F parameterizing the "double lines" in the threefold. We prove that this curve reconstructs V and hence we get a Torelli-like result: the infinitesimal invariant for the Fano cycle determines V.