Moduli map of second fundamental forms on a nonsingular intersection of two quadrics

Abstract

Griffiths and Harris (Ann Sci Ec Norm Super 12: 355-432, 1979) asked whether a projective complex submanifold of codimension two is determined by the moduli of its second fundamental forms. More precisely, given a nonsingular subvariety Xn. Pn+ 2, the second fundamental form IIX, x at a point x. X is a pencil of quadrics on Tx (X), defining a rational map mu X from X to a suitable moduli space of pencils of quadrics on a complex vector space of dimension n. The question raised by Griffiths and Harris was whether the image of mu X determines X. We study this question when Xn. Pn+ 2 is a nonsingular intersection of two quadric hypersurfaces of dimension n > 4. In this case, the second fundamental form IIX, x at a general point x. X is a nonsingular pencil of quadrics. Firstly, we prove that the moduli map mu X is dominant over the moduli of nonsingular pencils of quadrics. This gives a negative answer to Griffiths-Harris's question. To remedy the situation, we consider a refined version mu X of the moduli map mu X, which takes into account the infinitesimal information of mu X. Our main result is an affirmative answer in terms of the refined moduli map: we prove that the image of mu X determines X, among nonsingular intersections of two quadrics.