The topics
Varieties, the classical dictionary between ideals and subsets of affine and projective space.
Sheaves, the sheaf condition, stalks, sheafification, and exactness.
Schemes, \(\Spec\) and \(\Proj\), gluing, and how to recognise a scheme’s properties from a ring.
Morphisms, separated, proper, finite, flat, smooth, and the valuative criteria.
Sheaves of modules, quasicoherence, twisting, and line bundles.
Divisors, Weil against Cartier, class groups, and Picard groups.
Cohomology, Čech computations, Serre duality, Riemann–Roch.
Curves and surfaces, genus in its several senses, Riemann–Hurwitz, embeddings.
Toric varieties, the construction from a fan, the orbit-cone correspondence, and the fan and polytope criteria for smoothness, completeness, projectivity, divisor classes, and sections of line bundles.
Stacks and moduli, functors of points, the étale and fppf topologies, descent, Hilbert schemes, algebraic spaces and stacks.
Resources, references, problem sources, and worked-solution collections.
Sample syllabi, topic lists from qualifying and oral examination syllabi.
Dependencies
A scheme is a locally ringed space, so the scheme pages use sheaves of rings and their stalks. Sheaf cohomology is defined for sheaves of abelian groups on a topological space and computed on schemes for quasicoherent sheaves. The curves and surfaces pages use Riemann–Roch, Serre duality, and the intersection pairing defined through line bundles.