On a noetherian integral separated scheme \(X\) whose local rings are unique factorization domains, Weil and Cartier divisors agree and \(\Cl X\cong\Pic X\); a rational function \(f\in K(X)^\times\) has Weil divisor \(\operatorname{div}f=\sum_Yv_Y(f)\,Y\), the sum over prime divisors \(Y\) with valuation \(v_Y\). For a singular curve, \(\Pic\) is computed by comparison with the normalization: over an algebraically closed field \(k\), the nodal plane cubic has \(\Pic^0\cong k^\times\) and the cuspidal plane cubic has \(\Pic^0\cong k\).
Weil and Cartier, the two definitions, the degree map, and the normalization and excision sequences for class groups.
Line bundles and linear systems, the sheaf \(\OO(D)\), complete linear systems, base loci, and the map to projective space.
Positivity, ample and very ample divisors, the cohomological criterion for ampleness, and linear, algebraic, and numerical equivalence.