For a smooth curve \(C\) on a smooth projective surface \(X\), adjunction gives \(2g(C)-2=C\cdot(C+K_X)\), which computes the genus of a curve from the intersection pairing on \(X\).
Genus, arithmetic, geometric, and topological genus, their computation from plane models, Hilbert polynomials, and Riemann–Hurwitz, and the curves of genus zero.
Embeddings and Jacobians, when the canonical system embeds, and \(\Pic^0\) as an abelian variety.
Special divisors and the canonical map, Clifford’s theorem, gonality, and hyperelliptic curves.
Elliptic curves, the group law, the \(j\)-invariant, the Hasse invariant, and rational points.
Elliptic curves over $\CC$, the Weierstrass \(\wp\)-function, uniformisation by a lattice, the modular function, and complex multiplication.
Moduli of curves, coarse and fine moduli spaces, the dimension \(3g-3\), and automorphisms as the obstruction to a fine moduli space.
Curves in projective space, projection, curves on a quadric, and which pairs \((d,g)\) occur.
Intersection theory on surfaces, adjunction, Riemann–Roch, and the Hodge index theorem.
Blowups and the classification of surfaces, the blowup formulas, contractibility, the cubic surface, and Kodaira dimension.