Rationality has been a central topic in the field since the Lüroth problem was formulated at the end of the 18th century. The only rational curve is the projective line, and Castelnuovo's criterion settles the two-dimensional case. However, determining which varieties are rational in higher dimensions can be a challenging problem. In these lectures, I will overview some of the history of the problem and focus on two aspects related to rationality: birational rigidity and deformations of rational varieties. We will prove Iskovskikh-Manin's theorem on quartic threefolds and its generalization to higher dimensions, and Kontsevich-Tschinkel's specialization result on rationality. Along the way, we will also review some classical theorems on surfaces such as Noether-Castelnuovo's factorization theorem of Cremona transformations and Segre and Manin's theorems on rationality of cubic surfaces over non-closed fields.
Basic notions in algebraic geometry (at the level of Hartshorne)
The following is not supposed to be an exhaustive list of references. More references are provided in the lecture notes.
J. Alexander, On the factorization of Cremona plane transformations. Trans. Amer. Math. Soc., 17(1916), 295–300
A. Corti, Singularities of linear systems and 3-fold birational geometry, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 259–312
T. de Fernex, Birationally rigid hypersurfaces, Invent. Math. 192 (2013), 533-566. Erratum, Invent. Math. 203 (2016), 675-680
T. de Fernex, Fano hypersurfaces and their birational geometry, in "Automorphisms in Birational and Affine Geometry," Levico Terme, Italy, October 2012, Springer Proceedings in Mathematics & Statistics, Vol. 79, 2014
T. de Fernex and D. Fusi, Rationality in families of threefolds, Rend. Circ. Mat. Palermo 62 (2013), 127–135
T. de Fernex, L. Ein and M. Mustata, Multiplicities and log canonical thresholds, J. Algebraic Geom. 13 (2004), 603-615
B. Hassett, A. Pirutka and Y. Tschinkel, Stable rationality of quadric surface bundles over surfaces. Acta Math. 220 (2018), 341–365
J. Kollár, K. Smith,and A. Corti, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004.
J. Kollár, The rigidity theorem of Fano-Segre-Iskovskikh-Manin-Pukhlikov-Corti-Cheltsov-de Fernex-Ein-Mustata-Zhuang, in Birational Geometry of Hypersurfaces, Gargnano del Garda, Italy, 2018, Lecture Notes of the Unione Matematica Italiana, Springer International Publishing, 2019
M. Kontsevich and Y. Tschinkel, Specialization of birational types. Invent. Math. 217 (2019), 415–432