An Invitation to Modern Enumerative Geometry

Beschreibung

This book is based on a series of lectures given by the author at SISSA, Trieste, within the PhD courses Techniques in enumerative geometry (2019) and Localisation in enumerative geometry (2021). The goal of this book is to provide a gentle introduction, aimed mainly at graduate students, to the fast-growing subject of enumerative geometry and, more specifically, counting invariants in algebraic geometry. In addition to the more advanced techniques explained and applied in full detail to concrete calculations, the book contains the proofs of several background results, important for the foundations of the theory. In this respect, this text is conceived for PhD students or research “beginners” in the field of enumerative geometry or related areas. This book can be read as an introduction to Hilbert schemes and Quot schemes on 3-folds but also as an introduction to localisation formulae in enumerative geometry. It is meant to be accessible without a strong background in algebraic geometry; however, three appendices (one on deformation theory, one on intersection theory, one on virtual fundamental classes) are meant to help the reader dive deeper into the main material of the book and to make the text itself as self-contained as possible.

This book is based on a series of lectures given by the author at SISSA, Trieste, within the PhD courses Techniques in enumerative geometry (2019) and Localisation in enumerative geometry (2021). The goal of this book is to provide a gentle introduction, aimed mainly at graduate students, to the fast-growing subject of enumerative geometry and, more specifically, counting invariants in algebraic geometry. In addition to the more advanced techniques explained and applied in full detail to concrete calculations, the book contains the proofs of several background results, important for the foundations of the theory. In this respect, this text is conceived for PhD students or research “beginners” in the field of enumerative geometry or related areas. This book can be read as an introduction to Hilbert schemes and Quot schemes on 3-folds but also as an introduction to localisation formulae in enumerative geometry. It is meant to be accessible without a strong background in algebraic geometry; however, three appendices (one on deformation theory, one on intersection theory, one on virtual fundamental classes) are meant to help the reader dive deeper into the main material of the book and to make the text itself as self-contained as possible.

Autor*in

Andrea T. Ricolfi

Themen in »An Invitation to Modern Enumerative Geometry«

Torus localisation Hilbert scheme of points Moduli spaces of sheaves Deformation theory Curve counting Curves and sheaves on threefolds Donaldson-Thomas invariants Quot schemes Calabi-Yau varieties Equivariant cohomology Atiyah-Bott localisation Virtual localisation Virtual fundamental classes Counting invariants Quiver representations

Stimmen zu »An Invitation to Modern Enumerative Geometry«

“This book offers an introduction to modern enumerative geometry. As an active research area of algebraic geometry, enumerative geometry is closely linked with and shaped by many areas of mathematics and physics … . It provides a comprehensive and excellent resource for both classical and modern enumerative geometry.” (Zhenbo Qin, Mathematical Reviews, July, 2025) 

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