An invitation to quantum cohomology: Kontsevich's formula for rational plane curves
by Joachim Kock and Israel Vainsencherxii+159pp., No. 249 of Progress in Mathematics, Birkhäuser, 2006.
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An invitation to quantum cohomology: Kontsevich's formula for rational plane curvesby Joachim Kock and Israel Vainsencherxii+159pp., No. 249 of Progress in Mathematics, Birkhäuser, 2006.
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Emphasis is given throughout the exposition to examples, heuristic discussions, and simple applications of the basic tools to best convey the intuition behind the subject. The book demystifies these new quantum techniques by showing how they fit into classical algebraic geometry.
Some familiarity with basic algebraic geometry and elementary intersection theory is assumed. Each chapter concludes with some historical comments and an outline of key topics and themes as a guide for further study, followed by a collection of exercises that complement the material covered and reinforce computational skills. As such, the book is ideal for self-study, as a text for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory. The book will prove equally useful to graduate students in the classroom setting as to researchers in geometry and physics who wish to learn about the subject.
\begin{obs}
Observe that the stabilization of the family $\ov U_{0,4}
\times_{\ov M_{0,4}}\ov U_{0,4} \to \ov U_{0,4}$ is not simply the
blowup along the intersections of the sections as it was the case
for $n=3$. For $n\geq 4$, the projection map has singular fibres,
and it is necesssary also to blow up where the diagonal meets the
singular loci of the fibres (cf.~case-I stabilization). In fact
this blowup results in a smooth variety.
\end{obs}