no. 4
On cubics and quartics through a canonical curve
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 803-822

We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a grassmannian and a Flag variety respectively. Using G. Kempf's cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve

Classification : 14H60, 14H42 
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     title = {On cubics and quartics through a canonical curve},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     year = {2003},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
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Pauly, Christian. On cubics and quartics through a canonical curve. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 803-822. https://www.numdam.org/item/ASNSP_2003_5_2_4_803_0/

[ACGH] E. Arbarello - M. Cornalba - P. A. Griffiths - J. Harris, “Geometry of algebraic curves”, vol. 1, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985. | Zbl | MR

[BV] S. Brivio - A. Verra, The theta divisor of 𝒮𝒰 C (2,2d) is very ample if C is not hyperelliptic, Duke Math. J. 82 (1996), 503-552. | Zbl | MR

[DK] I. Dolgachev - V. Kanev, Polar Covariants of Plane Cubics and Quartics, Adv. Math. 98 (1993), 216-301. | Zbl | MR

[vGvG] B. Van Geemen - G. Van Der Geer, Kummer varieties and the moduli spaces of abelian varieties, Amer. J. Math. 108 (1986), 615-642. | Zbl | MR

[vGI] B. Van Geemen - E. Izadi, The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the Jacobian, J. Algebraic Geom. 10 (2001), 133-177. | Zbl | MR

[Gr] M. Green, Quadrics of rank four in the ideal of the canonical curve, Invent. Math. 75 (1984), 85-104. | Zbl | MR

[IP] E. Izadi - C. Pauly, Some Properties of Second Order Theta Functions on Prym Varieties, Math. Nachr. 230 (2001), 73-91. | Zbl | MR

[K1] G. Kempf, “Abelian Integrals”, Monografias Inst. Mat. No. 13, Univ. Nacional Autónoma Mexico, 1984. | Zbl | MR

[K2] G. Kempf, The equation defining a curve of genus 4, Proc. of the Amer. Math. Soc. 97 (1986), 214-225. | Zbl | MR

[KS] G. Kempf - F.-O. Schreyer, A Torelli theorem for osculating cones to the theta divisor, Compositio Math. 67 (1988), 343-353. | Numdam | Zbl | MR | EuDML

[OPP] W. M. Oxbury - C. Pauly - E. Previato, Subvarieties of 𝒮𝒰 C (2) and 2θ-divisors in the Jacobian, Trans. Amer. Math. Soc., Vol. 350 (1998), 3587-3614. | Zbl | MR

[PP] C. Pauly - E. Previato, Singularities of 2θ-divisors in the Jacobian, Bull. Soc. Math. France 129 (2001), 449-485. | Numdam | Zbl | MR | EuDML

[We] G. Welters, The surface C-C on Jacobi varieties and second order theta functions, Acta Math. 157 (1986), 1-22. | Zbl | MR