Unexpected hypersurfaces and where to find them

In a recent paper by Cook, et al., which introduced the concept of unexpected plane curves, the focus was on understanding the geometry of the curves themselves. Here we expand the definition to hypersurfaces of any dimension and, using constructions which appeal to algebra, geometry, representation theory and computation, we obtain a coarse but complete classification of unexpected hypersurfaces. In particular, we determine each $(n,d,m)$ for which there is some finite set of points $Z\subset\mathbb P^n$ with an unexpected hypersurface of degree $d$ in $\mathbb P^n$ having a general point $P$ of multiplicity $m$. Our constructions also give new insight into the interesting question of where to look for such $Z$. Recent work of Di Marca, Malara and Oneto \cite{DMO} and of Bauer, Malara, Szemberg and Szpond \cite{BMSS} give new results and examples in $\mathbb P^2$ and $\mathbb P^3$. We obtain our main results using a new construction of unexpected hypersurfaces involving cones. This method applies in $\mathbb P^n$ for $n \geq 3$ and gives a broad range of examples, which we link to certain failures of the Weak Lefschetz Property. We also give constructions using root systems, both in $\mathbb P^2$ and $\mathbb P^n$ for $n \geq 3$. Finally, we explain an observation of \cite{BMSS}, showing that the unexpected curves of \cite{CHMN} are in some sense dual to their tangent cones at their singular point.