This volume, "Geometry of Algebraic Curves II," is part of the "Grundlehren der mathematischen Wissenschaften" series and focuses on the foundations of the theory of moduli of algebraic curves defined over the complex numbers. The book is a comprehensive and detailed treatment of the subject, blending multiple perspectives—algebro-geometric, complex-analytic, topological, and combinatorial—used for the study of the moduli of curves. The content is structured around several key topics: 1. **Hilbert Scheme and Deformation Theory**: Chapters IX and XI introduce the Hilbert scheme and deformation theory, which are crucial for understanding the moduli space of stable curves. 2. **Moduli Space of Stable Curves**: Chapter XIII constructs the moduli space $\overline{M}_{g,n}$ of stable $n$-pointed curves of genus $g$. 3. **Line Bundles and Projectivity**: Chapter XIV discusses line bundles on moduli stacks and proves the projectivity of $\overline{M}_{g,n}$. 4. **Teichmüller Theory**: Chapter XV provides a self-contained treatment of Teichmüller space and modular groups. 5. **Smooth Galois Covers of Moduli Spaces**: Chapter XVI explores the theory of smooth Galois covers of moduli spaces. 6. **Cycles in Moduli Spaces**: Chapter XVII delves into the theory of cycles in $\overline{M}_{g,n}$. 7. **Cellular Decomposition**: Chapter XVIII introduces a combinatorial triangulation of Teichmüller space. 8. **Chow Rings and Intersection Theory**: Chapter XIX discusses the Chow rings and intersection theory of tautological classes. 9. **Kontsevich's Proof of Witten's Conjecture**: Chapter XX presents Kontsevich's combinatorial proof of Witten's conjecture on intersection numbers of $\psi$-classes. 10. **Brill–Noether Theory**: Chapter XXI studies Brill–Noether theory for smooth curves moving with moduli. The book also includes a guide for the reader, a list of symbols, and bibliographical notes, providing a comprehensive resource for researchers and students in the field of algebraic geometry.This volume, "Geometry of Algebraic Curves II," is part of the "Grundlehren der mathematischen Wissenschaften" series and focuses on the foundations of the theory of moduli of algebraic curves defined over the complex numbers. The book is a comprehensive and detailed treatment of the subject, blending multiple perspectives—algebro-geometric, complex-analytic, topological, and combinatorial—used for the study of the moduli of curves. The content is structured around several key topics: 1. **Hilbert Scheme and Deformation Theory**: Chapters IX and XI introduce the Hilbert scheme and deformation theory, which are crucial for understanding the moduli space of stable curves. 2. **Moduli Space of Stable Curves**: Chapter XIII constructs the moduli space $\overline{M}_{g,n}$ of stable $n$-pointed curves of genus $g$. 3. **Line Bundles and Projectivity**: Chapter XIV discusses line bundles on moduli stacks and proves the projectivity of $\overline{M}_{g,n}$. 4. **Teichmüller Theory**: Chapter XV provides a self-contained treatment of Teichmüller space and modular groups. 5. **Smooth Galois Covers of Moduli Spaces**: Chapter XVI explores the theory of smooth Galois covers of moduli spaces. 6. **Cycles in Moduli Spaces**: Chapter XVII delves into the theory of cycles in $\overline{M}_{g,n}$. 7. **Cellular Decomposition**: Chapter XVIII introduces a combinatorial triangulation of Teichmüller space. 8. **Chow Rings and Intersection Theory**: Chapter XIX discusses the Chow rings and intersection theory of tautological classes. 9. **Kontsevich's Proof of Witten's Conjecture**: Chapter XX presents Kontsevich's combinatorial proof of Witten's conjecture on intersection numbers of $\psi$-classes. 10. **Brill–Noether Theory**: Chapter XXI studies Brill–Noether theory for smooth curves moving with moduli. The book also includes a guide for the reader, a list of symbols, and bibliographical notes, providing a comprehensive resource for researchers and students in the field of algebraic geometry.