Projective normality of canonical symmetric squares

TL;DR

This paper extends classical results on projective normality from curves to their symmetric squares, establishing conditions under which the symmetric square's canonical bundle yields a projectively normal embedding.

Contribution

It proves an analogue of Noether's theorem for the symmetric square of a curve, identifying when the canonical bundle induces a projectively normal embedding.

Findings

The symmetric square's canonical bundle is projectively normal if and only if the curve is not hyperelliptic, trigonal, or a smooth plane quintic.

The result parallels classical theorems for curves, extending them to symmetric squares.

The theorem characterizes the ideal generation by quadrics in the context of symmetric squares.

Abstract

Recall that a smooth complex projective curve has a very ample canonical bundle when it is non-hyperelliptic, and according to a theorem of M. Noether the resulting embedding is projectively normal. A theorem of Petri further asserts that the homogeneous ideal is generated by quadrics if the curve is neither trigonal nor a smooth plane quintic. In this note, we prove an analogue of Noether's theorem for the symmetric square of the curve - namely, the canonical bundle of the symmetric square determines a projectively normal embedding exactly when the curve itself is neither hyperelliptic, trigonal nor a smooth plane quintic. The theorem of Petri highlights the governing role played by quadric generation of the ideal of the curve.