Spaces of conics on low degree complete intersections

Abstract

Let X X be a smooth complete intersection contained in P C n \mathbb {P}^n_{\mathbb {C}} and of low degree. We consider conics contained in X X and passing through two general points of X X . We show that the moduli space of these conics is a smooth complete intersection in a projective space. The main ingredients of the proof are a criterion for characterizing when a smooth projective variety is a complete intersection in a projective space, the Grothendieck-Riemann-Roch theorem, and the geometry of spaces of conics.