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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.
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