# New constructions of unexpected hypersurfaces in $\mathbb{P}^n$

## TL;DR

This paper introduces new methods for constructing unexpected hypersurfaces in projective spaces, expanding the known classes by using cones on positive dimensional varieties and birational transformations.

## Contribution

It presents three novel approaches to generate unexpected varieties, including cones on higher-dimensional varieties and birational transformations for higher-dimensional hypersurfaces.

## Key findings

- Cones on positive dimensional varieties of codimension ≥ 2 often produce unexpected hypersurfaces.
- Constructed unexpected surfaces from lines in P^3.
- Generalized constructions to higher dimensions using birational transformations.

## Abstract

In the paper we present new examples of unexpected varieties. The research on unexpected varieties started with a paper of Cook II, Harbourne, Migliore and Nagel and was continued in the paper of Harbourne, Migliore, Nagel and Teitler. Here we present three ways of producing unexpected varieties that expand on what was previously known. In the paper of Harbourne, Migliore, Nagel and Teitler, cones on varieties of codimension 2 were used to produce unexpected hypersurfaces. Here we show that cones on positive dimensional varieties of codimension 2 or more almost always give unexpected hypersurfaces. For non-cones, almost all previous work has been for unexpected hypersurfaces coming from finite sets of points. Here we construct unexpected surfaces coming from lines in $\mathbb{P}^3$, and we generalize the construction using birational transformations to obtain unexpected hypersurfaces in higher dimensions.