Let k be an algebraically closed field and let C be a nonhyperelliptic smooth projective curve of genus g defined over k. Since the canonical model of C is arithmetically Gorenstein, Macaulay’s theory of inverse systems allows us to associate to C a cubic form f in the divided power k-algebra Rg−3 in g − 2 variables. The apolarity ap(C) of C is the minimal number t of linear form in Rg−3 needed to write f as the sum of their divided power cubes. It is easy to see that ap(C) is at least g − 2 and P. De Poi and F. Zucconi classified curves with ap(C) = g −2 when k is the complex number field. In this paper, we give a complete, characteristic free, classification of curves C with apolarity g −1 (and g −2).

Canonical curves with low apolarity / Ballico, Edoardo; G., Casnati; R., Notari. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 332:1(2011), pp. 229-243. [10.1016/j.jalgebra.2010.12.030]

Canonical curves with low apolarity

Ballico, Edoardo;
2011-01-01

Abstract

Let k be an algebraically closed field and let C be a nonhyperelliptic smooth projective curve of genus g defined over k. Since the canonical model of C is arithmetically Gorenstein, Macaulay’s theory of inverse systems allows us to associate to C a cubic form f in the divided power k-algebra Rg−3 in g − 2 variables. The apolarity ap(C) of C is the minimal number t of linear form in Rg−3 needed to write f as the sum of their divided power cubes. It is easy to see that ap(C) is at least g − 2 and P. De Poi and F. Zucconi classified curves with ap(C) = g −2 when k is the complex number field. In this paper, we give a complete, characteristic free, classification of curves C with apolarity g −1 (and g −2).
2011
1
Ballico, Edoardo; G., Casnati; R., Notari
Canonical curves with low apolarity / Ballico, Edoardo; G., Casnati; R., Notari. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 332:1(2011), pp. 229-243. [10.1016/j.jalgebra.2010.12.030]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/86323
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