Unexpected hypersurfaces of type $(d+k,d)$

TL;DR

This work extends the phenomenon of unexpected hypersurfaces from planes to higher dimensions by developing a syzygy-based construction that produces hypersurfaces of type $$ (d+k,d) $$ in $\mathbb{P}^n$, vanishing along a general codimension-two subspace with multiplicity $d$. Central to the method is a correspondence between $k$-derivations of the dual hyperplane arrangement $\mathcal{A}_Z$ and the graded module $I(Z,\mathcal{H})(-k)$, realized via the restriction $D_0^k(\mathcal{A}_Z)|_{\mathcal{L}}$ and the map $\eta$, which yields a hypersurface $\mathcal{S}$ of degree $\le d+k$ through $Z$ with the prescribed multiplicity along $\mathcal{H}$. A numerical criterion for unexpectedness is given in terms of the splitting type of $D_0^k(\mathcal{A}_Z)$ along the dual line $\mathcal{L}$, enabling explicit checks and the discovery of higher-dimensional examples, including cases in $\mathbb{P}^4$ and $\mathbb{P}^3$. The approach unifies planar results with higher-dimensional constructions and demonstrates how syzygies and higher-order derivations govern unexpectedness across dimensions. The work also highlights phenomena such as multiplicities arising at fixed configuration points, illustrated by concrete crystallographic and Fermat configurations.

Abstract

Unexpected hypersurfaces arise when vanishing in points of a set $Z$ and higher-order vanishing along a general linear subspace fails to impose the expected number of independent conditions on forms of a fixed degree. The phenomenon was first observed for planar curves by Cook, Harbourne, Migliore and Nagel. This paper shows a syzygy-based construction of, possibly unexpected, hypersurfaces of degree $d+k$ in $\mathbb{P}^n$, vanishing along a codimension two general linear subspace with multiplicity $d$; thus generalizing the work of Trok and the previous work of the last two authors. Our framework unifies the classical planar cases with higher-dimensional examples, including Trok's construction. We give a sufficient criterion for unexpectedness (via the splitting behaviour the syzygy bundles of the powers of the Jacobian ideal, associated with the hyperplane arrangement dual to $Z$) and provide explicit examples in $\mathbb{P}^3$ and $\mathbb{P}^4$.

Theorems & Definitions (33)

• Definition 1.1: HMT-G
• Definition 2.1
• Theorem 3.1
• proof
• Definition 4.1
• Lemma 4.2
• proof
• Remark 4.3
• Proposition 4.4
• proof