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

TL;DR

This paper introduces a syzygy-based method to construct and analyze unexpected hypersurfaces in projective spaces, generalizing previous planar cases to higher dimensions and providing criteria for their unexpectedness.

Contribution

It develops a unifying framework for constructing unexpected hypersurfaces using syzygies, extending known results from planar curves to higher-dimensional projective spaces.

Findings

Provides a syzygy-based construction method for unexpected hypersurfaces.

Unifies classical planar cases with higher-dimensional examples.

Offers a criterion for unexpectedness based on syzygy bundle splitting.

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 $\mathbb{P}^4$.