Abstract
This survey retraces the author’s talk at the Workshop Birational geometry of surfaces, Rome, January 11–15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces.
Similar content being viewed by others
Change history
08 January 2018
The purpose of this note is to fix an imprecision in Theorem 3.3 of the original paper, which affects the correctness of the statement and leads to a mistake
References
Alzati, A., Pirola, G.P.: On the holomorphic length of a complex projective variety. Arch. Math. (Basel) 59, 392–402 (1992)
Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 3(25), 75–95 (1972)
Barth, W.: Abelian surfaces with \((1,2)\)-polarization. In: Oda, T. (ed.) Algebraic Geometry, Sendai, 1985, pp. 41–84, Adv. Stud. Pure Math., vol. 10. North-Holland, Amsterdam (1987)
Bastianelli, F., Pirola, G.P., Stoppino, L.: Galois closure and Lagrangian varieties. Adv. Math. 225, 3463–3501 (2010)
Bastianelli, F.: On symmetric products of curves. Trans. Am. Math. Soc. 364, 2493–2519 (2012)
Bastianelli, F., Cortini, R., De Poi, P.: The gonality theorem of Noether for hypersurfaces. J. Algebraic Geom. 23, 313–339 (2014)
Bastianelli, F.: On irrationality of surfaces in \(\mathbb{P}^3\). arXiv:1603.05543 (2016)
Bastianelli, F., De Poi, P., Ein, L., Lazarsfeld, R., Ullery, B.: Measures of irrationality for hypersurfaces of large degree. arXiv:1511.01359 (2016)
Bastianelli, F., Ciliberto, C., Flamini, F., Supino, P.: A note on gonality of curves on general hypersurfaces (2016) (preprint)
Beauville, A.: Complex Algebraic Surfaces, 2nd edn. London Mathematical Society Student Texts, vol. 34. Cambridge University Press, Cambridge (1996)
Beauville, A., Colliot-Thélène, J.L., Sansuc, J.J., Swinnerton-Dyer, P.: Variétés stablement rationnelles non rationnelles. Ann. Math. 121, 283–318 (1985)
Birkenhake, C., Lange, H.: Complex Abelian Varieties, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302. Springer, Berlin (2004)
Chen, X.: Rational curves on K3 surfaces. J. Algebraic Geom. 8, 245–278 (1999)
Ciliberto, C.: Alcune applicazioni di un classico procedimento di Castelnuovo. Seminari di geometria. 1982–1983 (Bologna, 1982/1983), pp. 17–43. Univ. Stud. Bologna, Bologna (1984)
Ciliberto, C., Knutsen, A.L.: On k-gonal loci in Severi varieties on general K3 surfaces and rational curves on hyperkhler manifolds. J. Math. Pures Appl. 101, 473–494 (2014)
Clemens, C.H., Griffiths, P.A.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281–356 (1972)
Cortini, R.: Il grado di irrazionalità di varietà algebriche, Ph.D. thesis. Università degli Studi di Genova, Genova (1999)
de Fernex, T., Fusi, D.: Rationality in families of threefolds. Rend. Circ. Mat. Palermo 62, 127–135 (2013)
Dolgačev, I.V.: Special algebraic \(K3\)-surfaces. Math. URRS Izvestija 7, 833–846 (1973)
Guerra, L., Pirola, G.P.: On rational maps from a general surface in \(\mathbb{P}^3\) to surfaces of general type. Adv. Geom. 8, 289–307 (2008)
Iskovskih, V.A., Manin, JuI: Three-dimensional quartics and counterexamples to the Lüroth problem. Mat. Sbornik 86, 140–166 (1971)
Knutsen, A.L.: Remarks on families of singular curves with hyperelliptic normalizations. Indag. Math. 19, 217–238 (2008)
Lopez, A.F., Pirola, G.P.: On the curves through a general point of a smooth surface in \(\mathbb{P}^3\). Math. Z. 219, 93–106 (1995)
Mumford, D.: Rational equivalence of \(0\)-cycles on surfaces. J. Math. Kyoto Univ. 9, 195–204 (1969)
Moh, T.T., Heinzer, W.: A generalized Lüroth theorem for curves. J. Math. Soc. Japan 31, 85–86 (1979)
Moh, T.T., Heinzer, W.: On the Lüroth semigroup and Weierstrass canonical divisor. J. Algebra 77, 62–73 (1982)
Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. In: Raynaud, M., Shioda, T. (eds.) Algebraic Geometry (Tokyo/Kyoto, 1982), pp. 334–353, Lecture Notes in Mathematics, vol. 1016. Springer, Berlin (1983)
Tokunaga, H., Yoshihara, H.: Degree of irrationality of abelian surfaces. J. Algebra 174, 1111–1121 (1995)
Yoshihara, H.: Degree of irrationality of an algebraic surface. J. Algebra 67, 634–640 (1994)
Yoshihara, H.: Quotients of abelian surfaces. Publ. Res. Inst. Math. Sci. 31, 135–143 (1995)
Yoshihara, H.: Degree of irrationality of a product of two elliptic curves. Proc. Am. Math. Soc. 124, 1371–1375 (1996)
Yoshihara, H.: Existence of curves of genus three on a product of two elliptic curves. J. Math. Soc. Japan 49, 531–537 (1997)
Yoshihara, H.: A note on the inequality of degrees of irrationalities of algebraic surfaces. J. Algebra 207, 272–275 (1998)
Yoshihara, H.: Degree of irrationality of hyperelliptic surfaces. Algebra Colloq. 7, 319–328 (2000)
Xu, G.: Subvarieties of general hypersurfaces in projective spaces. J. Differ. Geom. 39, 581–589 (1994)
Acknowledgements
I would like to thank the Organizers of the Workshop Birational geometry of surfaces and Andrea Bruno for encouraging me to think about this topic in view of the Workshop. I am grateful to Gian Pietro Pirola for introducing me to these problems when I was a Ph.D. student. I would also like to thank Andreas Knutsen for valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant agreement no. 307119, MIUR FIRB 2012 “Spazi di moduli e applicazioni”, MIUR PRIN 2010–2011“Geometria delle varietà algebriche”, and INdAM (GNSAGA).
Rights and permissions
About this article
Cite this article
Bastianelli, F. Irrationality issues for projective surfaces. Boll Unione Mat Ital 11, 13–25 (2018). https://doi.org/10.1007/s40574-017-0116-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40574-017-0116-2