Obstructions to deforming space curves lying on a complete intersection of quadrics

Source abstract

Let $S=S_{2,\ldots,2}\subset\mathbb P^{n+2}$ be a smooth complete intersection of $n$ quadrics containing a line $E$, where $n\geq 3$, and set $D=(n-1)\mathbf h+E$, where $\mathbf h=[\mathcal O_S(1)]$. We study the deformations of smooth curves $C\in |mD|$ and their obstructions. The linear system $|D|$ defines a birational morphism $f:S\to\overline S$ that contracts $E$ to a point and realizes an unprojection in the sense of Papadakis--Reid. When $n=3$, the surface $S=S_{2,2,2}\subset\mathbb P^5$ is a $K3$ surface and $E$ is a $(-2)$-curve. We prove that the complementary open surface $S\setminus E$ admits first-order infinitesimal deformations in $\mathbb P^5\setminus E$ that are obstructed at the second order. Using these obstructions, we show that, for every sufficiently large $m$ with $\operatorname{char} k\nmid m$, there exists a generically non-reduced irreducible component $W_m$ of the Hilbert scheme $\operatorname{Hilb}^{sc}\mathbb P^5$ of smooth connected curves in $\mathbb P^5$. Its general member $C$ lies on a complete intersection $S_{2,2,2}$ of three quadrics, satisfies $C\sim mD$, and has $h^0(N_{C/\mathbb P^5})-\dim_{[C]}\operatorname{Hilb}^{sc}\mathbb P^5=3$. In contrast, when $n>3$, we prove that, for every sufficiently large $m$ with $\operatorname{char} k\nmid m$, the corresponding family of curves in $|mD|$ gives rise to a generically smooth irreducible component of $\operatorname{Hilb}^{sc}\mathbb P^{n+2}$.