Unexpected curves and Togliatti-type surfaces

TL;DR

This paper links unexpected hypersurfaces to varieties with defective osculating spaces, identifying specific plane curves as sections of a rational surface with lower-than-expected osculating space dimensions, and relates this to classical Togliatti surfaces.

Contribution

It establishes a direct connection between unexpected hypersurfaces and defective osculating behavior, providing new insights into Togliatti-type surfaces and their properties.

Findings

Identification of unexpected plane curves of degree 4 as sections of a rational surface in P^5.

Demonstration of defective osculating spaces in these surfaces.

Analysis of Lefschetz properties related to the surfaces studied.

Abstract

The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of degree 7 in P^5 with its osculating spaces of order 2 which in every point of X have dimension lower than expected. We put this result in perspective with earlier examples of surfaces with defective osculating spaces due to Shifrin and Togliatti. Our considerations are rendered by an analysis of Lefschetz Properties of ideals associated with the studied surfaces.