Wall-crossing on universal compactified Jacobians

Enumerative geometry is one of the most ancient fields of mathematics, and it aims at counting the number of geometric objects having a certain property. For example, we may ask how many straight lines pass through two given points in the plane. It is Euclid's very first axiom that asserts that there is a unique such line. Another example is to count how many points belong simultaneously to two lines in the plane. Here Euclid's fifth axiom essentially implies that the answer is one if and only if the lines are not parallel. For a slightly more interesting example, one could consider a parabola and a circle in the plane, and see that the number of points belonging to both could be any number between 0 and 4 (depending on the relative position of the line and the circle.

The examples above hopefully demonstrate how such questions can be basic and pervasive in geometry, and they give a glimpse onto geometry's early historical developments. Today the field is still existing and very active, and it employs techniques coming from different fields of mathematics. In the last 25 years, revolutionary ideas in the field have arrived from physics, in particular from theories originating from the quest of unifying the four fundamental forces, like string theory.

The main modern approach to counting theories today uses moduli spaces. How many quadrics pass through 5 general points in the plane? A possible approach is to consider the 5-dimensional (projective) space that parametrizes plane quadrics, and to realize that the constraint of passing through a point corresponds to cutting a hyperplane in such space. By intersecting the 5 hyperplanes, we find out that the answer to the counting question is 1.

The proposed research follows this paradigm to approach some questions in algebraic geometry. The moduli spaces studied in this proposal are moduli of line bundles of some fixed degree over projective algebraic curves, and the constraints are given (for example) by imposing that such line bundles have a given number of linearly independent global sections (Brill-Noether loci). In order to construct such moduli spaces one has to introduce an "extra" parameter, not a-priori imposed by the problem of parameterizing the aforementioned geometric objects, called stability. This parameter is a continuous parameter, but the moduli space actually varies only when the parameter crosses some hyperplanes (called walls) in the space where it lives.

Our point of view is that the geometric picture should simplify when one considers all stability parameters, rather than only one. For example, there is usually one "easy" parameter, for which the given constraints and their geometric nature can be easily understood and there is one "interesting" parameter that has received lots of attention from several mathematicians. The novelty of our approach consists in finding results for the moduli space corresponding to the "interesting" parameter by first solving the same problem for the "easy" parameter, and then investigating how the moduli spaces vary with the stability parameter when a wall is crossed. The different moduli spaces should be related to each other by flips (and going into a wall should correspond to a contraction).

The proposed research and the success in the various objectives will result in several research papers. All the preprints will be made available via the arXiv, and new updates will be timely uploaded. The papers will be submitted for publication to prestigious mathematical journals to ensure maximal visibility from the community. The PI will base his decision using the ArticleInfluence Score, a metric for journals that is based on the same principles of the page ranking algorithms for web searches, implemented by ISI Web of Knowledge (Thomson Reuters). The PI prefers publishing on general mathematical journals rather than on specialized ones. When writing research papers, the PI enjoys working to make the introduction useful and readable to the widest possible audience.

This research will impact through participation in conferences and seminars, research visits, the organization of one workshops half-way through the proposed research, and by means of outreach/dissemination targeted at younger mathematicians and high-school pupils.