## Final Activity Report Summary - CALABI-YAU VARIETIES (Calabi-Yau varieties and Mori fiber spaces)

Classification theory is both one of the most classical fields in algebraic geometry, dating back to the Italian school of the 19th century, and at the same time a very important and fast developing field in today's algebraic geometry, with applications ranging for instance to the very important question in physics of classifying and studying so-called Calabi-Yau varieties, which represent the part of the universe that has been compactified after the Big Bang.

The classical way to study the classification problem is to introduce numerical invariants, study their possible values and then look at families of geometrical objects, called varieties, with such invariants. However this method gets more and more difficult as the dimension grows, already for dimension 3. Therefore, a lot of interest has recently been devoted to finding methods to recognise a variety from its subvarieties, i.e. subobjects of lower dimension. In this context a very important contribution in the past decades is the approach proposed by fields medallist Shigefumi Mori. The idea is to contract subvarieties until reaching a simpler object to classify.

The MC project aimed at finding and applying an approach to classification theory that is in particular applicable to two special outcomes of Mori's program, namely Mori fibre spaces and Calabi-Yau varieties. The idea is to subsequently take sections of the variety until one reaches curves and study certain classical properties of those to obtain some information about the variety one started with.

The most important achievement made was an optimal bound on an invariant of a particular type of Mori fibre spaces called Enriques-Fano threefolds. These varieties have dimension three and have the property that its general surface section is an Enriques surface, a surface that has been studied classically. EF threefolds were claimed classified by Fano in 1938 but his proof contains several gaps. The new approach was to study properties of certain families of curves covering a general Enriques surface section and deduce that if the properties were fulfilled, then the surface couldn't be a section of a threefold. This gave the desired bound on the invariant in question. To study the properties of the curves, four more papers concerning curves on Enriques surfaces were written. These deal with concrete geometrical properties of curves on Enriques surfaces.

Families of rational curves on special Calabi-Yau varieties of dimension 4, called symplectic fourfolds, were also studied. Rational curves are in many ways the simplest possible curves, in that they enjoy the same properties as straight lines. In physics they represent the simplest interactions between particles, in Mori's program they govern which varieties should be contracted, and in symplectic fourfolds they are known to enjoy many more important properties, so that one can safely say that those are the most important curves to study. The achievement of the joint work is to construct several families of previously undetected rational curves, as well as obtain some more general results about these curves and their connection with curves on K3 surfaces. As a spin-off product new results about curves on K3 surfaces are obtained.

Finally, 3 more papers concerning curves on surfaces with the researcher were produced:

one studies singular curves (roughly speaking, curves with edges) on surfaces with the special property that a desingularisation has gonality 2 (the gonality is the invariant of a curve that is the smallest degree of a folding of the curve onto a rational curve);

the second studies curves on K3 surfaces that are highly singular (that is, with high multiplicity) at one point;

the third is concerned with a 20 year old conjecture about curves on K3 surfaces. The latter stated that curves moving in a family on a K3 surface should have the same gonality. One counterexample was known, and the scientist shows that this is the only counterexample.

The classical way to study the classification problem is to introduce numerical invariants, study their possible values and then look at families of geometrical objects, called varieties, with such invariants. However this method gets more and more difficult as the dimension grows, already for dimension 3. Therefore, a lot of interest has recently been devoted to finding methods to recognise a variety from its subvarieties, i.e. subobjects of lower dimension. In this context a very important contribution in the past decades is the approach proposed by fields medallist Shigefumi Mori. The idea is to contract subvarieties until reaching a simpler object to classify.

The MC project aimed at finding and applying an approach to classification theory that is in particular applicable to two special outcomes of Mori's program, namely Mori fibre spaces and Calabi-Yau varieties. The idea is to subsequently take sections of the variety until one reaches curves and study certain classical properties of those to obtain some information about the variety one started with.

The most important achievement made was an optimal bound on an invariant of a particular type of Mori fibre spaces called Enriques-Fano threefolds. These varieties have dimension three and have the property that its general surface section is an Enriques surface, a surface that has been studied classically. EF threefolds were claimed classified by Fano in 1938 but his proof contains several gaps. The new approach was to study properties of certain families of curves covering a general Enriques surface section and deduce that if the properties were fulfilled, then the surface couldn't be a section of a threefold. This gave the desired bound on the invariant in question. To study the properties of the curves, four more papers concerning curves on Enriques surfaces were written. These deal with concrete geometrical properties of curves on Enriques surfaces.

Families of rational curves on special Calabi-Yau varieties of dimension 4, called symplectic fourfolds, were also studied. Rational curves are in many ways the simplest possible curves, in that they enjoy the same properties as straight lines. In physics they represent the simplest interactions between particles, in Mori's program they govern which varieties should be contracted, and in symplectic fourfolds they are known to enjoy many more important properties, so that one can safely say that those are the most important curves to study. The achievement of the joint work is to construct several families of previously undetected rational curves, as well as obtain some more general results about these curves and their connection with curves on K3 surfaces. As a spin-off product new results about curves on K3 surfaces are obtained.

Finally, 3 more papers concerning curves on surfaces with the researcher were produced:

one studies singular curves (roughly speaking, curves with edges) on surfaces with the special property that a desingularisation has gonality 2 (the gonality is the invariant of a curve that is the smallest degree of a folding of the curve onto a rational curve);

the second studies curves on K3 surfaces that are highly singular (that is, with high multiplicity) at one point;

the third is concerned with a 20 year old conjecture about curves on K3 surfaces. The latter stated that curves moving in a family on a K3 surface should have the same gonality. One counterexample was known, and the scientist shows that this is the only counterexample.