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

Abstract

In [GH], Griffiths and Harris 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 $II_{X,x}$ at a point $x \in X$ is a pencil of quadrics on $T_x(X)$, defining a rational map $μ^X$ 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 $μ^X$ 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 $II_{X,x}$ at a general point $x \in X$ is a nonsingular pencil of quadrics. Firstly, we prove that the moduli map $μ^X$ 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 $\widetildeμ^X$ of the moduli map $μ^X$, which takes into account the infinitesimal information of $μ^X$. Our main result is an affirmative answer in terms of the refined moduli map: we prove that the image of $\widetildeμ^X$ determines $X$, among nonsingular intersections of two quadrics.