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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.

##### MSC:
 14H20 Singularities of curves, local rings 14H45 Special algebraic curves and curves of low genus 14H51 Special divisors on curves (gonality, Brill-Noether theory)
##### Keywords:
canonical model; singular curve; non-Gorenstein curve
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##### References:
 [1] Arbarello E., Cornalba M., Griffiths P.A., Harris J.: Geometry of Algebraic Curves. Springer- Verlag, Berlin (1985) · Zbl 0559.14017 [2] Bertini E.: Introduzione alla Geometria Proiettiva degli Iperspazi. Enrico Spoerri, Pisa (1907) · JFM 38.0582.02 [3] Barucci V., Fröberg R.: One-dimensional almost Gorenstein rings. J. Alg 188, 418–442 (1997) · Zbl 0874.13018 · doi:10.1006/jabr.1996.6837 [4] Catanese, F., Pluricanonical-Gorenstein-curves, In Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., 24, Birkhäuser Boston, 1982, pp. 51–95 [5] Eisenbud D.: Linear sections of determinantal varieties. Am. J. Math. 110(3), 541–575 (1988) · Zbl 0681.14028 · doi:10.2307/2374622 [6] Eisenbud, D.: Commutative algebra with a view towards algebraic geometry. Graduate Texts in Mathematics 150. Springer, New York (1994) [7] Eisenbud, D., Koh, J., Stillman, M. (appendix with Harris, J.): Determinantal equations for curves of high degree. Am. J. Math. 110, 513–539 (1988) · Zbl 0681.14027 [8] Fujita, T.: Defining equations for certain types of polarized varieties. In Complex Analysis and Algebraic Geometry, pp. 165–173. Iwanami Shoten, Tokyo (1977) · Zbl 0353.14011 [9] Fujita T.: On hyperelliptic polarized varieties. Thoku Math. J. 35(2), 1–44 (1983) · Zbl 0505.14003 · doi:10.2748/tmj/1178229099 [10] Grothendieck, A., and Dieudonné, J.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. vol. 8 (1961) [11] Hartshorne R.: Algebraic Geometry. Springer-Verlag, New York (1977) · Zbl 0367.14001 [12] Homma M.: Singular hyperelliptic curves. Manuscr. Math. 98, 21–36 (1999) · Zbl 0940.14019 · doi:10.1007/s002290050122 [13] Hartshorne R.: Generalized divisors on Gorenstein curves and a theorem of Noether. J. Math. Kyoto Univ. 26-3, 375–386 (1986) · Zbl 0613.14008 [14] Kempf, G.: The singularities of certain varieties in the Jacobian of a curve. PhD thesis, Columbia University (1971) [15] Kleiman S., Landolfi J.: Geometry and deformation of special Schubert varieties. Compos. Math. 23, 407–434 (1971) · Zbl 0238.14006 [16] Lipman J.: Stable ideals and Arf rings. Am. J. Math. 93(3), 649–685 (1971) · Zbl 0228.13008 · doi:10.2307/2373463 [17] Martins R.V.: On trigonal non-Gorenstein curves with zero Maroni invariant. J. Algebra 275, 453–470 (2004) · Zbl 1060.14036 · doi:10.1016/j.jalgebra.2003.10.033 [18] Mumford, D.: Lectures on curves on an algebraic surface. With a section by G.M. Bergman, Annals of Mathematics Studies, No. 59. Princeton University Press, Princeton (1966) · Zbl 0187.42701 [19] Rosenlicht M.: Equivalence relations on algebraic curves. Ann. Math. 56, 169–191 (1952) · Zbl 0047.14503 · doi:10.2307/1969773 [20] Saint-Donat B.: On Petri’s analysis of the linear system of quadrics through a canonical curve. Math. Ann. 206, 157–175 (1973) · Zbl 0315.14010 · doi:10.1007/BF01430982 [21] Serre J.P.: Groupes Algébriques et Corps de Classes. Hermann, Paris (1959) · Zbl 0097.35604 [22] Stöhr K.-O.: On the poles of regular differentials of singular curves. Bull. Braz. Math. Soc. 24, 105–135 (1993) · Zbl 0788.14020 · doi:10.1007/BF01231698 [23] Stöhr K.-O.: Hyperelliptic Gorenstein curves. J. Pure Appl. Algebra 135, 93–105 (1999) · Zbl 0940.14018 · doi:10.1016/S0022-4049(97)00124-2
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