On the Severi problem

TL;DR: In this article, the authors present notes from a jointly taught class at the University of Chicago and lectures by the first author in Santa Cruz, discussing moduli spaces of stable maps, Gromov-Witten invariants, quantum cohomology, and examples.

TL;DR: In this paper, the authors obtained a recursive formula answering the following question: How many irreducible, plane curves of degree d and genus g pass through 3d-1+g general points in the plane?

TL;DR: The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems and those who are not interested in explicit machine computations at all.

TL;DR: In this article, the degeneration of linear series on smooth curves is studied, as the curves degenerate to a certain type of reducible curves, curves of compact type, called limit linear series.

TL;DR: Severi and Anhang as discussed by the authors showed that the dimension of a complete (i.e., maximal) irreducible algebraic system, defined over k, of plane algebraic curves of a given order n, such that the general curve C* of En d /k has d nodes (and no other singularities), is equal to 3n + p 1, where p = ((n 1)(n 2)/2) d is the "effective" genus of C*.

TL;DR: From the combination of knowledge and actions, someone can improve their skill and ability and this will lead them to live and work much better.