On unexpected curves of type $(d+k,d)$

TL;DR

This paper introduces a construction for unexpected algebraic curves of degree d+k passing through a point set Z with a generic point P of multiplicity d, using syzygies of Jacobian powers and bundle splitting types.

Contribution

It provides a novel geometric construction and a characterization of unexpected curves via syzygy bundle splitting types, advancing understanding of their algebraic and geometric properties.

Findings

Construction explains existence of unexpected curves of degree d+k

Characterization of unexpectedness via splitting type of syzygy bundles

Provides new tools for analyzing algebraic curves with prescribed singularities

Abstract

We present a construction explaining the existence of (unexpected) curves of degree $d+k$, passing through a set $Z$ of points on $\mathbb{P}^2$, and having a generic point $P$ of multiplicity $d$. The construction is based on the syzygies of the $k$-th powers of Jacobian of the product of lines dual to the points of $Z$. We prove also a result characterizing the unexpectedness of the curves via splitting type of the bundle of these syzygies retricted to the line dual to $P$.