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arXiv · 2211.08313

Projective normality of canonical symmetric squares

John Sheridan
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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.

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John Sheridan. 2022-11-15. Projective normality of canonical symmetric squares. https://arxiv.org/abs/2211.08313

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