Invitation to Modern Enumerative Geometry

Andrea T. Ricolfi obtained a PhD in Mathematics at the University of Stavanger (Norway) and is now an assistant professor at SISSA, Trieste. His research interests include the study of moduli spaces of sheaves on algebraic 3-folds and refined invariants attached to them, as well as derived categories of coherent sheaves, quiver representations and Grothendieck rings of varieties.

• 1 Introduction.- 2 Counting in algebraic geometry.- 3 Background material.- 4 Informal introduction to Grassmannians.- 5 Relative Grassmannians, Quot, Hilb.- 6 The Hilbert scheme of points.- 7 Equivariant Cohomology.- 8 The Atiyah-Bott localisation formula.- 9 Applications of the localisation formula.- 10 The toy model for the virtual fundamental class and its localization.- 11 Degree 0 DT invariants of a local Calabi-Yau 3-fold.- 12 DT/PT correspondence and a glimpse of Gromov–Witten theory .- Appendix A: Deformation Theory.- Appendix B: Intersection Theory.- Appendix C: Perfect obstruction theories and virtual classes.

“The book under review is intended to be a gentle introduction to contemporary enumerative geometry. … The book is intended to be suitable for young graduate students and to be self-contained enough to be read in one piece. … I can definitely recommend it as a rather advanced invitation to enumerative geometry.” (Piotr Pokora, zbMATH 1508.14001, 2023)