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# Families of Subvarieties in Complex Algebraic Varieties

## Periodic Reporting for period 1 - FOSICAV (Families of Subvarieties in Complex Algebraic Varieties)

"This is a project in complex algebraic geometry: A 'variety' is an object defined by polynomial equations in the space with coordinates in the field of complex numbers. One fundamental aspect of algebraic geometry is that varieties vary in families, and that these families ('moduli spaces') are themselves varieties.

The central theme is the geometric study of various families of subvarieties in some prescribed varieties. For example, take S a surface, L a class of polynomial equations on S, and g an integer; the 'Severi variety' V_L^g(S) is the family of curves of genus g in S
defined by an equation of class L. (Complex algebraic curves may be seen as Riemann surfaces; the 'genus' of a Riemann surface is its ""number of holes"").

Of particular interest are the 'enumerative properties' of these families. For example, plane curves of degree 3 and genus 0 (these are necessarily singular) move in a family of dimension 8; fix 8 points in the projective plane; how many curves in the family are there that contain all 8 points? (answer: 12). This kind of numbers defined on a variety V are interesting per se for the algebraic geometer, but are also important invariants attached to V (Gromov-Witten invariants). They play a prominent role in theoretical physics, specifically in string theory; the most relevant case is when V is a Calabi-Yau variety of dimension 3.

We focus on 'K3 surfaces'. They are the surfaces S with zero curvature that are simply connected (the latter property means that every loop on S may be deformed continously to a loop of length 0). Surfaces defined by one single degree 4 equation in 3-space (quartic surfaces) are K3.

Our approach is by 'degeneration': to study a family F of subvarieties in V, we let V degenerate, then try to understand the limit of F, and get information on F back from this. In the 90s Ran, Caporaso-Harris, Vakil, studied the Severi varieties of the plane by degeneration to the union of two surfaces; they set up a rule (RCHV) for the limits of Severi varieties in this situation. A key objective for us was to show the existence for any degeneration of well-behaved surfaces (eg, K3) of a 'good model', suitable for the description of limit Severi varieties in terms of the RCHV rule. Our guiding example was the degeneration of a quartic K3 to a tetrahedron. This is much more complicated than in the RCHV situation, if only because of 'triple points', at which three irreducible component of the tetrahedron meet."
1) We have built a good model for the degeneration of K3 surfaces in 5-space to an octahedron

This is a further experimentation, after our work [CD14] on the tetrahedron before the project. Unexpectedly, interesting new phenomena with respect to [CD14] showed up, and we made important discoveries toward our general objective of building good models in general. The difficulties we encoutered on our way were rather formidable, and we have not yet reached our general objective.

Two secondments have been set up, which led to decisive progress. The first one with Brugallé about the tropical aspects of both enumerative geometry and the degeneration techniques. This helped us shape an original view on degenerate K3 surfaces and the limit linear systems therein (see pictures). The second one with Abramovich and Chen, about their very recent work with Gross and Siebert, in which they use logarithmic geometry to derive a Gromov-Witten theoretic degeneration formula valid for virtually any degeneration once it is put in logarithmically smooth form. The ideas therein will be of great help to reach our objective in the near future.

2) Degeneration of singular hypersurfaces and application to enumeration (with Galati)

This is twofold: i) find a higher dimensional analog of the RCHV rule; ii) set up a degeneration strategy to apply this rule to concrete calculations.

We have obtained a first version of the desired rule, using limit linear series techniques à la Ciliberto-Miranda. The complete analysis is is still work in progress. Although this rule parallels rather faithfully the RCHV one, the shift in dimension requires a dramatic change in techniques for the proofs.

We have elaborated a prototypical application to enumerative geometry, resolving the first new difficulties that show up as the dimension increases.

3) Let (S,L) be a very general primitively polarized K3 surface of genus p. For all positive integers g such that g>=2p/3, the Severi variety V_L^g (S) is irreducible.

Virtually nothing general was known before on this question.

4) Degenerations of Enriques surfaces and applications (with Galati and Knutsen)

Although Enriques surfaces are close cousins of K3 surfaces, their geometry is really special. We have built two semi-stable surfaces, and shown that they are well-behaved degenerations of Enriques surfaces.

This already gave a description by generators and relations of the group of divisors modulo numerical equivalence of a general Enriques surface. Various other applications are in progress: i) the determination of the limit of the effective cone under the two degenerations; ii) the study of the modular map from the universal curve over the moduli space of polarized Enriques surfaces to the moduli space of curves; and iii) an enumerative study of the Severi varieties of Enriques surfaces.

5) Wahl maps and extensions of canonical curves and K3 surfaces (with Sernesi)

Those curves whose canonical embedding is a hyperplane section of a K3 surface have been characterized by Wahl and Arbarello-Bruno-Sernesi by the non-surjectivity of their so-called 'Wahl maps'. We characterize more generally the curves for which the Wahl map has a given corank r as those which can be extended to a (1+r)-dimensional variety. As a byproduct, we obtain a characterization of those K3 surfaces that are hyperplane section of a Fano threefold.
We expect our down-to-earth approach to enumerative invariants to enable computations by nature out of reach of the Gromov-Witten arsenal: typically, the number of rational curves of class kL, k>=2, on a K3 surface. There is a formula giving the corresponding Gromov-Witten invariant, but this number does not correspond to the number of curves we are interested in.

This is a project in pure mathematics. As such, it was not an objective to seek for any *direct* application. It is a well documented fact that pure science fuels more applied research, and eventually the development of concrete applications. The particular research under report may or may not have applications in the future, there is no way we or anyone can tell right now. We believe it is another task to look for immediate practical applications, and we leave it to someone more competent than us.

As a matter of fact, we strongly support the idea that research in pure science should be sustained regardless of its possible applications, because it helps shaping our way of thinking: consider how the concept of truth has evolved through scientific changes. Moreover science provides perspective to mankind, and it is an achievement of civilization to create enough wealth to be able to sustain non-profit art, science, culture.