13/2 ways of counting curves (Q2875833)

The paper under review is an excellent survey on the subject of counting curves, written by two leading experts in the field. A curve counting theory from the modern viewpoint, loosely speaking, involves a geometrically meaningful compactification of the families of curves in an algebraic variety \(X\) as well as (almost always) a deformation/obstruction theory (to define a virtual fundamental class). The authors summarize major ways of counting curves in the last two decades, including naive counting (classical enumerative geometry), Gromov-Written theory, Gopakumar-Vafa/BPS invariants, Donaldson-Thomas theory, stable pairs, stable unramified maps and stable quotients, with a focus on the case when \(X\) is a nonsingular projective \(3\)-fold. Moreover, the authors analyze in detail the advantages, drawbacks, and relationships amongst these approaches. This paper is well written and can serve as a great reference for students and general mathematicians looking for an elementary route into the subject.NEWLINENEWLINEAs a remark, the fractional part \(1/2\) in the title reflects the authors' opinion that naive curve counting is not always well-defined and has many drawbacks, so it should be viewed as only \(1/2\) a method. This adds an amusing flavor to the paper, which coincides with the fact that sometimes counting invariants are rational numbers only. It also implies that many topics discussed in the paper are still developing and await further discovery.NEWLINENEWLINEFor the entire collection see [Zbl 1286.14002].