A note on special cubic fourfolds of small discriminants

In this paper, our purpose is to give a characterization of the generic special cubic fourfold which contains a smooth rational surface of degree 9 not homologous to a complete intersection. As corollaries, we will give an explicit construction of families of smooth surfaces in generic special cubic fourfolds X ∈ 𝒞 δ {X\in\mathcal{C}_{\delta}} for 6 < δ ≤ 30 {6<\delta\leq 30} and δ ≡ 0 ( mod 6 ) {\delta\equiv 0~{}(\bmod~{}6)} . This applies in particular to give an explicit construction of two different liaison class of smooth surfaces in all such special cubic fourfolds with the prescribed invariants.

Schemes Supported on the Singular Locus of a Hyperplane Arrangement in $\mathbb P^n$

A hyperplane arrangement in $\mathbb P^n$ is free if $R/J$ is Cohen–Macaulay (CM), where $R = k[x_0,\dots ,x_n]$ and $J$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $ J^{un}$, the intersection of height two primary components, and $\sqrt{J}$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $\mathbb P^3$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $r$, there is an arrangement for which $R/J^{un}$ (resp. $R/\sqrt{J}$) fails to be CM in only one degree, and this failure is by $r$. We draw consequences for the even liaison class of $J^{un}$ or $\sqrt{J}$.