Abstract
Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such collections of varieties: their minimal degree and their maximal number of linearly independent smallest degree hypersurfaces passing through them. We show results for curves and surfaces, and pose several questions.
Similar content being viewed by others
References
Alzati, A., Russo, F.: On the \(k\)-normality of projected algebraic varieties. Bull. Braz. Math Soc, New Ser. 33(1), 27–48 (2007)
Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves, volume I, grundlehren der mathematischen wissenschaften, vol. 267. Springer-Verlag, New York (1985)
Ballico, E., Bolondi, G., Ellia, P., Mirò-Roig, R.M.: Curves of maximum genus in the range A and stick-figures. Trans. Amer. Math. Soc. 349(11), 4589–4608 (1997)
Ballico, E., Ellia, P.: On projections of ruled and Veronese varieties. J. Algebra 121, 477–487 (1989)
Ballico, E., Ellia, P.: On postulation of curves in \(\mathbb{P}^4\). Math. Z. 188, 215–223 (1985)
Ballico, E., Ellia, P.: The maximal rank conjecture for non-special curves in \({ {P}}^3\). Invent. Math. 79, 541–555 (1985)
Ballico, E., Ellia, P.: Beyond the maximal rank conjecture for curves in \({\mathbb{P}}^3\). In: Space curves, proceedings Rocca di Papa, pp. 1–23, Lecture Notes in Math. 1266, Springer, Berlin, (1985)
Ballico, E., Ellia, P.: The maximal rank conjecture for non-special curves in \({{\mathbb{P}}}^n\). Math. Z. 196, 355–367 (1987)
Ballico, E., Ellia, P.: The maximal genus of space curves in the range A, preprint at arXiv:1811.08807, (2018)
Ballico, E., Ellia, P., Fontanari, C.: Maximal rank of space curves in the range A. Eur. J. Math. 4, 778–801 (2018)
Brodmann, M., Schenzel, P.: Arithmetic properties of projective varieties of almost minimal degree. J. Algebraic Geom. 16, 347–400 (2007)
Chiantini, L., Ciliberto, C.: Towards a Halphen theory of linear series on curves. Trans. Amer. Math. Soc. 351(6), 2197–2212 (1999)
Di Gennaro, V., Franco, D.: Refining Castelnuovo-Halphen bounds. Rend. Circ. Mat. Palermo (2) 61(1), 91–106 (2012)
Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88(1), 89–133 (1984)
Fløystad, G.: Construction of space curves with good properties. Math. Ann. 289(1), 33–54 (1991)
Fløystad, G.: On space curves with good cohomological properties. Math. Ann. 291(3), 505–549 (1991)
Fujisawa, T.: On non-rational numerical Del Pezzo surfaces. Osaka J. Math. 32, 613–636 (1995)
Gruson, L., Peskine, C.: Genre des courbes de l’espace projectif. (French) Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977), pp. 31–59, Lecture Notes in Math., 687, Springer, Berlin (1978)
Gruson, L., Lazarsfeld, R., Peskine, C.: On a theorem of Castelnuovo and the equations defining space curves. Inv. Math. 72, 491–506 (1983)
Harris, J.: A bound on the geometric genus of projective varieties. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 8(1), 35–68 (1981)
Hartshorne, R.: Algebraic Geometry. Springer-Verlag, Berlin-Heidelberg-New York (1977)
Hartshorne, R.: On the classification of algebraic space curves. In: Vector bundles and differential equations (Nice 1979), p. 82–112, Progress in Math. 7 Birkhäuser, Boston (1980)
Hartshorne, R., Hirschowitz, A.: Nouvelles courbes de bon genre dans l’espace projectif. Math. Ann. 280(3), 353–367 (1988)
Hirschowitz, A.: Sur la postulation générique des courbes rationnelles. Acta Math. 146, 209–230 (1981)
Hoa, L.-T., Stückrad, J., Vogel, W.: Towards a structure theory for projective varieties of degree \(=\) codimension \(+ 2\). J. Pure Appl. Algebra 71, 203–231 (1991)
Lazarsfeld, R.: A sharp Castelnuovo bound for smooth surfaces. Duke Math. J. 55, 423–429 (1987)
L’vovsky, S.: On the inflection points, monomial curves, and hypersurfaces containing projective curves. Math. Ann. 306, 719–735 (1996)
Macrì, E., Schmidt, B.: Derived categories and the genus of space curves, preprint at arXiv:1801.02709, (2018)
Matsuki, K.: Introduction to the Mori program. Springer, Berlin (2002)
Park, E.: Smooth varieties of almost minimal degree. J. Algebra 314, 185–208 (2007)
Park, E.: On hypersurfaces containing projective varieties. Forum Math. 27(2), 843–875 (2015)
Szpiro, L.: Lectures on Equations defining Space Curves. Tata Institute of Fundamental Research, Bombay (1979)
Acknowledgements
The first author was partially supported by MIUR and GNSAGA of INdAM (Italy). The second author would like to thank the Department of Mathematics of Università di Trento, where part of this project was conducted, for the warm hospitality. We are grateful to an anonymous referee for pointing out a mistake in an earlier version and providing useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ballico, E., Ventura, E. Minimal degree equations for curves and surfaces (variations on a theme of Halphen). Beitr Algebra Geom 61, 297–315 (2020). https://doi.org/10.1007/s13366-019-00471-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-019-00471-w