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

Abstract

Griffiths and Harris (Ann Sci Ec Norm Supér 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 $X^n \subset {\mathbb {P}}^{n+2}$, the second fundamental form $\textit{II}_{X,x}$ at a point $x \in X$ is a pencil of quadrics on $T_x(X)$, 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 $X^n \subset {\mathbb {P}}^{n+2}$ is a nonsingular intersection of two quadric hypersurfaces of dimension $n >4$. In this case, the second fundamental form $\textit{II}_{X,x}$ at a general point $x \in X$ 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 $\widetilde{\mu }^X$ 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 $\widetilde{\mu }^X$ determines X, among nonsingular intersections of two quadrics.