Enumerative geometry is a branch of algebraic geometry which counts a number of solutions to various geometric problems, usually stated for data in general position. Usually the solution is a nonnegative integer, but some vaguely defined intuitive enumerative problems have rigorous reformulations in terms of integration on moduli spaces and can yield answers which are, say, rational numbers. Most studied enumerative problems are problems of intersection theory of algebraic subvarieties and the solution is often found by cohomological methods.
How many plane conics are tangent simultaneously to 5 given plane conics in general position ? The answer is 3264, see Griffiths-Harris, Principles of algebraic geometry for a detailed treatment. A further question is how many of them are real ? see Frank Sottile, 3264 real curves.
Wikipedia, Enumerative geometry
William Fulton, Introduction to intersection theory in algebraic geometry, CBMS 54, AMS, 1996, second edition
M. Kontsevich, Enumeration of rational curves via torus actions, arXiv:hep-th/9405035.
G. Ellingsrud, S. A. Strømme], Bott’s formula and enumerative geometry, JAMS vol. 9, num. 1, 1996, arXiv:alg-geom/9411005.
In relation to string theory (mostly Gromov-Witten theory):
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