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

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 $P$.