curvecounting

Curvecounting is a field within algebraic geometry that deals with enumerating geometric objects, specifically rational curves, that satisfy certain geometric conditions. The primary goal is to count how many such curves pass through a given set of points or are tangent to a given set of lines, or more generally, satisfy more complex incidence conditions. This is often achieved by developing techniques to define and calculate characteristic classes of vector bundles associated with these curves.

The foundation of curvecounting can be traced back to the work of mathematicians like Chasles and Cayley in the 19th century, who studied problems related to counting lines tangent to surfaces. Modern developments in the field are deeply intertwined with the theory of moduli spaces, which provide a framework for parameterizing families of curves. The Gromov-Witten invariants, a key concept introduced by Witten, have revolutionized curvecounting by connecting it to quantum field theory and providing a powerful computational tool.

The challenge in curvecounting lies in the fact that the objects being counted are often not discrete but live in continuous families. This necessitates the use of sophisticated tools from differential geometry and algebraic topology to handle these continuous aspects and ensure that the counts are well-defined and invariant under small deformations of the problem. The field has seen significant progress with the development of techniques like degeneration formulas and virtual moduli spaces, which allow for the calculation of invariants in more general situations. Curvecounting has found applications in various areas, including string theory, mirror symmetry, and the study of singularity theory.