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# Wall-Crossing and Algebraic Geometry

## Periodic Reporting for period 3 - WallCrossAG (Wall-Crossing and Algebraic Geometry)

Periodo di rendicontazione: 2022-06-01 al 2023-11-30

The goal of this project is to answer fundamental questions about the geometry of algebraic varieties. Algebraic varieties can be given as the solution set of a system of polynomial equations. Despite fast-moving progress that has completely changed the field several times over the last half century, some of the oldest problems remain among the most challenging: how to detect that two varieties are different, or how to detect that they are different in an essential way.

Algebraic varieties arise naturally whenever points in space (for example, data points in a high-dimensional space of measurements) are related by polynomials. They have found applications in statistics, in phylogenetics, in robotics, and many other fields. Many of the fundamental questions of algebraic geometry (for example, whether we can switch between implicit descriptions of geometric sets via equations, or explicit descriptions via a parametric form), have direct consequences in these applications.

The methodology of this project is based on wall-crossing and Bridgeland stability conditions, originally developed with a view towards string theory and counting invariants. Recent developments have turned them into a potentially widely applicable tool to enhance basic techniques and methods in algebraic geometry. The goal of this project is to bring these methods to their full potential, and apply them to a broad range of questions.

The project will enhance the underlying methodology - e.g. by the construction of stability conditions in higher dimensions. But it will also apply them to yield tangible progress on algebraic geometry questions of classical flavour, for example in Brill-Noether theory or the geometry of higher-dimensional Fano varieties. This is leading to fundamental progress on some of the most fundamental questions of algebraic geometry: the classification of algebraic varieties, whether they admit parametric forms, etc.
Joint work of the PI yielded a notion of stability conditions suitable when we are given an entire family of algebraic varieties, rather than a single one. This allows us to combine the notion of stability condition with the long-standing ubiquitous algebraic geometry technique of deforming and degenerating algebraic varieties. This is transformative methodology, that has already been applied in many surprising contexts.

The construction of stability conditions on higher-dimensional varieties has made significant progress, in work by team member Naoki Koseki. This is a key tool that has already been applied by a group at Imperial College to deduce fundamental consequences for numbers of curves on three-dimensional spaces.
The circle of ideas behind this work has also led to much stronger versions of the classical Bogomolov-Gieseker inequality in a wide range of situations, and to the first systematic construction of stability conditions in finite characteristic.

In work by various team members, much progress has been made in understanding the geometry of Fano varieties and their derived categories, including in the singular case. The fundamental question at hand here is that of the classification of algebraic varieties of a certain fixed type: what does the parameters space of all such varieties look like? Which invariants can reliably detect that two such varieties are distinct? On a technical level, results by various team members have greatly clarified the role of certain categories, called "Kuznetsov components", can play in the classification. These are now understood to be linked intrinsically to classical Hodge-theoretic classification data, but give much more powerful algebraic and geometric tool for the reconstruction of a given variety from its associated invariants.
In joint work in progress by the PI, a new approach has been developed towards the question which cubic fourfolds admit parametrisations, i.e. which of them are "rational". This will undoubtedly lead to significant progress; how far this idea will go is still to be seen.
Much further progress can be expected towards the classification of three-dimensional Fano varieties, for example a more complete understanding of the parameter space of Gushel-Mukai threefolds.