The Canonical Model of a Singular Curve

TL;DR

This paper refines and modernizes Rosenlicht's classical results on the canonical model of complete integral curves, establishing key properties and conditions for various normality types, with new characterizations for when the canonical model is rational normal.

Contribution

It provides modern proofs and new criteria for the properties of the canonical model of singular curves, including conditions for birational equivalence and normality types.

Findings

C and C' are birationally equivalent iff C is nonhyperelliptic

C' equals the blowup of C with respect to the canonical sheaf if C is nonhyperelliptic

Criteria for C' being rational normal, arithmetically normal, projectively normal, and linearly normal

Abstract

We give refined statements and modern proofs of Rosenlicht's results about the canonical model C' of an arbitrary complete integral curve C. Notably, we prove that C and C' are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C' is equal to the blowup of C with respect to the canonical sheaf \omega. We also prove some new results: we determine just when C' is rational normal, arithmetically normal, projectively normal, and linearly normal.