Enumerative Geometry: from Classical to Modern

Yu-Jong Tzeng (Harvard)

The subject of enumerative geometry goes back at least to the middle of the 19th century. It studies the number of geometric objects in a given class that satisfy a number of incidence or tangency conditions. For example, (a) How many lines pass through distinct 2 points? (b) How many lines pass through 4 general lines in 3-space? (c) How many rational cubics pass through 9 points on the projective plane? The field has been very active in the last twenty years due to its interaction with physics, which provides motivation for many theories and conjectural formulas. One of the most famous problem in enumerative geometry is computing Severi degrees, which are the numbers of degree d plane curves with a given number of nodes and pass through an appropriate number of points in general position. The computation of Severi degrees, as well as counting nodal curves on other surfaces are great examples of classical problems which were open for long time and can finally be solved by modern techniques now. Furthermore, a recent conjecture of Gottsche states that there should be a universal formula tha t computes the number of nodal curves in a sufficiently ample linear system on any surface. In this talk I will first give an indication of what enumerative geometry is about, then give a survey about counting curves on several surfaces. Finally I will explain why Gottsche's conjecture is true and discuss its generalizations.

Tuesday, January 15th, at 4:00pm in HBH 227

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