Quartic unexpected curves and surfaces

TL;DR

This paper explores unexpected algebraic curves and surfaces arising from special point configurations, revealing new properties in projective spaces and extending known phenomena to higher dimensions.

Contribution

It introduces novel unexpected surfaces in three-dimensional projective space, expanding the understanding of such phenomena beyond previously studied configurations.

Findings

Identification of new properties of nine-point configurations in the plane.

Construction of a surface in three-space with unexpected postulation properties.

Extension of unexpected algebraic phenomena to higher dimensions.

Abstract

Our research is motivated by recent work of Cook II, Harbourne, Migliore, and Nagel on configurations of points in the projective plane with properties that are unexpected from the point of view of the postulation theory. In this note, we revisit the basic configuration of nine points appearing in work of Gennaro/Ilardi/Vall\`es and Harbourne, and we exhibit some additional new properties of this configuration. We then pass to projective three-space and exhibit a surface with unexpected postulation properties there. Such higher dimensional phenomena have not been observed so far.