FOSICAV · Families of Subvarieties in Complex Algebraic Varieties

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."

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