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The canonical model of a singular curve. (English) Zbl 1172.14019
Let \(C\) be a complete integral curve of arithmetic genus \(g \geq 2\) over an algebraically closed field of arbitrary characteristic. The canonical model \(C'\) of \(C\) was defined by M. Rosenlicht [Ann. Math. (2) 56, 169–191 (1952 (1952; Zbl 0047.14503)], where he introduced the dualizing sheaf \(\omega\).
In this paper the authors give refined statements and modern proofs of Rosenlicht’s results about \(C'\):
They give a version of Clifford’s Theorem more general with respect to Rosenlicht’s one and they characterize the case in which the canonical model \(C'\) is equal to the rational normal curve \(N_{g-1}\).
Then they show that, if \(C\) is nonhyperelliptic, the canonical map induces an open embedding of its Gorenstein locus into its canonical model \(C'\); the proof involves the blowup of \(C\) with respect to \(\omega\).
Also, using Castelnuovo Theory and some results due to V. Barucci and R. Fröberg [J. Algebra 188, No. 2, 418–442 (1997; Zbl 0874.13018)], they give necessary and sufficient conditions for the canonical model \(C'\) to be arithmetically normal.
Finally, they prove Rosenlicht’s Main Theorem which essentially asserts that, if \(C\) is nonhyperelliptic, the canonical map between the blowup of \(C\) with respect to \(\omega\) and \(C'\) is an isomorphism; they apply this result to characterize the non-Gorenstein curves \(C\) whose canonical model is projectively normal.

14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus
14H51 Special divisors on curves (gonality, Brill-Noether theory)
Full Text: DOI
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