Birational geometry of foliations on surfaces

Topics

In the Fall Semester 2023, the reading is organized by Luca Tasin (with the collaboration of Roberto Svaldi).
We will read on the birational classification of foliations on surfaces.

The main goal will be to understand the classification that was carried out at the turn of the 20th century by, mainly, McQuillan, Brunella, and Mendes and that grew out of McQuillan's proof of the Green-Griffiths Conjecture for general type surfaces satisfying certain numerical constraints.
Time permetting, we will try to get an introduction to recent techniques, problems, and results for foliations on higher dimensional varieties (and rank higher than 1).
The reading course, at least ofr the part concerning foliations on surfaces, will loosely follow the now-classical reference book Birational geometry of foliations by Marco Brunella, which we shall indicate with [Bru] below.

Lectures and calendar

To see a tentative calendar and list of topics for the reading seminar just click here.
If you are interested in giving one of the talks, please contact Luca or Roberto.

Here is the list of lectures together with the materials:

1. 23.10.2023. Roberto Svaldi: Introduction and motivations for the study of the birational geometry of foliations.
   Topics:
   • definition of foliation;
   • some basic examples;
   • motivations for studying foliations: the Green-Griffiths Conjecture and McQuillan's proof in the case of surfaces; Miyaoka's Generic Semipositivity Theorem; destabilizing subsheaves of the tangent bundle and the proof of the Abundance Conjecture in dimension three;
   • analogies with the Minimal Model Program and the classification of surfaces;
   • foliated singularities, resolution (or, rather lack thereof), and Seidenberg's theorem;
   • the foliated MMP for foliated surfaces and the current state of the art.
   Notes: [Lecture 1, 1st part] [Lecture 1, 2nd part]
   Some further papers to read for the topics mentioned in the talk:
   • M. Brunella, Courbes entières dans les surfaces algébriques complexes, Séminaire Bourbaki : volume 2000/2001, exposés 880-893, Astérisque, no. 282 (2002), Exposé no. 881, 23 p. [In french]
   • N. Shepherd-Barron, Miyaoka's Theorem on the generic semipositivity of \( T_X \) and on the Kodaira dimension of minimal regular threefolds, Chapter 9 of Flips and abundance for algebraic threefolds (Salt Lake City, UT, 1991) Astérisque(1992), no.211, pp. 1–258.


2. 30.10.2023. Luca Tasin: Basics on the structure of foliations on surfaces.
   Topics:
   • the exact sequence of a foliation on a smooth surface and equivalent definitions using vector fields and differential forms;
   • local examples;
   • multiplicity of singularities and formula to compute the global multiplicity, see [Bru, Proposition 2.1];
   • invariant curves and adjunction formulas, see [Bru, Propositions 2.2-3];
   • global examples: fibrations and foliations on the projective plane, cf. [Bru, 2.3].
   Notes: [Lecture 2]


3. 13.11.2023. Saverio Secci: Singularities of foliations and blow-ups.
   Topics:
   • definition of reduced singularity and some remarks on their geometry;
   • constuction of the induced foliation on the blow-up of a surface at a point and its properties;
   • Seidenberg's theorem [Bru, thm.1.1];
   • definition of dicritical singularities and their characterization via blow-ups [Bru, prop.1.1];
   • remarks on the blow-up along non-reduced singularities;
   • examples: foliations associated to the 1-forms \( xdy+ydx, xdy-ydx, dx \) on \( \mathbb C^2 \).
   Notes: [Lecture 3]


4. 27.11.2023. Priyankur Chaudhuri: Minimal models of foliated surfaces.
   Topics:
   • definition and properties of Riccati foliations;
   • foliated exceptional curves;
   • relatively minimal and minimal foliations: definitions and basic properties;
   • examples of foliations which are relatively minimal but not minimal.
   Notes: [Lecture 4]


5. 04.12.2023. Luigi Lombardi: A rationality criterion using foliations.
   Topics:
   • foliation in positive characteristic;
   • p-closed foliations;
   • Miyaoka’s theorem: statement and proof.
   Notes: [Lecture 5]


6. 11.12.2023. Priyankur Chaudhuri: surface foliations without minimal models.
   Topics:
   • definition of the very special foliation [Bru, Chapter 4, section 2];
   • birational classification of foliations without minimal model [Bru, Theorem 5.1].
   Notes: [Lecture 6]


7. 15.01.2024. Luca Tasin: classification of foliated surfaces according to their Kodaira dimension.
   Topics:
   • definition of Kodaira and numerical dimension for a divisor on a surface;
   • structure theorem for the negative part in the Zariski decomposizion of \( K_{\mathcal F} \) (Theorem 8.1, without proof);
   • structure theorem for relatively minimal fibrations of numerical dimension 0 (Theorem 8.2);
   • summary of the classification of foliations generated by global vector fields with isolated zeros.
   Notes: [Lecture 7]


8. 30.01.2024. Roberto Svaldi: index theorem and their applications.
   Topics:
   • definition of the Baum-Bott residue of the Baum-Bott formula (Theorem 3.1);
   • definition of the Camacho-Sad index and of the variation index;
   • Camacho-Sad formula (Theorem 3.2);
   • existence of separatrices (Theorem 3.3) and sketch of the proof.
   Notes: [Lecture 8]
9. 20.02.2024. Priyankur Chaudhuri: concluding the classification of foliated surfaces.
   Topics:
   • full classification of foliations of Kodaira dimension 1, [Bru, Theorem 9.1];
   • Kodaira dimension 0 is equivalente to numerical dimension 0 for minimal foliated surface, [Bru, Theorem 9.2];
   • discussion of the cases in which abundance does not hold for foliated surfaces.
   Notes: [Lecture 9]