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

Let $S=S_{2,\dots ,2}\subset {\mathbb{P}}^{n+2}$ be a smooth complete intersection of $n$ quadrics containing a line $E$, where $n\ge 3$, and set $D=(n-1)\mathbf{h}+E$, where $\mathbf{h}=[{\mathcal{O}}_S(1)]$. We study the deformations of smooth curves $C\in \mathrm{\mid }mD\mathrm{\mid }$ and their obstructions. The linear system $\mathrm{\mid }D\mathrm{\mid }$ defines a birational morphism $f:S\to \bar{S}$ 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 $\mathrm{char}k\nmid m$, there exists a generically non-reduced irreducible component $W_m$ of the Hilbert scheme ${\mathrm{Hilb}}^{sc}{\mathbb{P}}^5$ 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 $h^0(N_{C\mathrm{/}{\mathbb{P}}^5})-{\dim }_{[C]}{\mathrm{Hilb}}^{sc}{\mathbb{P}}^5=3$. In contrast, when $n>3$, we prove that, for every sufficiently large $m$ with $\mathrm{char}k\nmid m$, the corresponding family of curves in $\mathrm{\mid }mD\mathrm{\mid }$ gives rise to a generically smooth irreducible component of ${\mathrm{Hilb}}^{sc}{\mathbb{P}}^{n+2}$.