Measures of irrationality for projective varieties (Lecture 1)

An important problem in algebraic geometry is understanding whether a given variety satisfies some rationality property, such as being  rational, uniruled, rationally connected, unirational or stably rational.

The purpose of these lectures is to investigate a complementary circle of questions: in what manner can one quantify and control `how irrational’ a given complex projective variety X might be?

In this direction, we will consider various birational invariants called `measures of irrationality’, which somehow measure the failure of a given projective

variety to satisfy the rationality properties listed above. In particular, I will focus on the `degree of irrationality’ (i.e. the least degree of a dominant rational map from X to the projective space) and the `covering gonality’ (i.e. the least gonality of a curves passing through a general point of X).

Initially, I will introduce these ideas and I will discuss various examples and results. Then I will present some techniques based on positivity properties of canonical bundles, which lead to lower bounds for those birational invariants. Finally, I will show how these techniques can be used to describe the invariants for hypersurfaces of large degree and for other varieties of general type.

– F. Bastianelli, R. Cortini and P. De Poi, The gonality theorem of Noether for hypersurfaces, J. Algebraic Geom. 23 (2014), 313-339.

– F. Bastianelli, P. De Poi, L. Ein, R. Lazarsfeld, B. Ullery, Measures of irrationality for hypersurfaces of large degree, Compos. Math. 153 (2017), 2368-2393.

– F. Bastianelli, Irrationality issues for projective surfaces, Boll. Unione Mat. Ital. 11 (2018), 13-25.

– F. Bastianelli, C. Ciliberto, F. Flamini, P. Supino, Gonality of curves on general hypersurfaces, J. Math. Pures Appl. 125 (2019), 94-118.