Topic lists from algebraic geometry qualifying and oral examination syllabi. Past examinations with transcribed questions are linked from Resources.
Harvard qualifying examination
Affine and projective varieties; regular functions and maps; cones and projections.
Projective space and the Grassmannian.
Ideals of varieties; the Nullstellensatz.
Rational functions, rational maps and blowing up.
Dimension and degree of a variety; the Hilbert function and Hilbert polynomial.
Smooth and singular points of varieties; the Zariski tangent space; tangent cones; dual varieties.
Families of varieties: Chow varieties and Hilbert schemes.
Algebraic curves: genus; the genus formula for plane curves; the Riemann–Hurwitz formula; the Riemann–Roch theorem.
References: Shafarevich, Basic Algebraic Geometry 1, 2nd edition; [Harr10].
Scheme theory, cohomology and curves
This syllabus accompanies a qualifying exam transcript.
Scheme theory:
Schemes, morphisms between schemes, separated and proper morphisms, the valuative criterion.
Quasicoherent and coherent sheaves, invertible sheaves, Weil divisors, Cartier divisors, the divisor class group, the Picard group.
Projective morphisms, blowing up, Kähler differentials, nonsingular varieties, Bertini’s theorem.
Cohomology:
Sheaf cohomology, Grothendieck vanishing, cohomology of Noetherian affine schemes.
Serre’s affineness criterion, Čech cohomology, cohomology of projective space.
The cohomological criterion for ampleness, Serre duality (statement), higher direct images.
Flat morphisms, flat families, Hilbert polynomials, smooth morphisms.
Curves:
The Riemann–Roch theorem, Hurwitz’s theorem, linear systems on curves, embeddings of curves in \(\PP^n\).
Elliptic curves (up to the group structure), hyperelliptic curves, canonical embeddings.
Clifford’s theorem, Castelnuovo’s theorem (statement), classification of low-degree curves in \(\PP^3\).
References: [Hart10].
Schemes and cohomology, by Hartshorne section
Schemes:
Sheaves: definition, presheaves and associated sheaves, morphisms, stalks, pullback and pushforward.
Schemes: affine schemes, \(\Proj\), the structure sheaf, the scheme associated to a variety.
First properties of schemes: reduced, integral and Noetherian schemes, morphisms of finite type, finite morphisms, closed immersions, dimension, fibre products.
Separated and proper morphisms: definitions, valuative criteria, projective morphisms, reduced structure on closed subsets, scheme-theoretic image, constructible sets.
Coherent and quasicoherent sheaves: \(\OO_X\)-modules, quasicoherent and coherent sheaves, constructions on \(\OO_X\)-modules, invertible sheaves, vector bundles.
Divisors: Weil divisors, Cartier divisors, their equivalence for locally factorial schemes, invertible sheaves.
Projective morphisms: criteria and characterization of projective morphisms, ample and very ample line bundles, blowups.
Differentials: derivations, the module of relative differential forms, sheaves of differentials and nonsingularity, the tangent sheaf, the canonical sheaf, geometric genus.
Cohomology:
Derived functors: abelian categories, complexes, derived functors, \(\delta\)-functors.
Cohomology of sheaves: the category of sheaves of \(\OO_X\)-modules has enough injectives; basic vanishing theorems.
Cohomology of a Noetherian affine scheme: vanishing of higher cohomology of quasicoherent sheaves, and Serre’s criterion for affineness in terms of vanishing of cohomology.
Čech cohomology: definition, and its isomorphism with derived-functor cohomology for a Noetherian separated scheme.
Cohomology of projective space: calculation by Čech cohomology, Serre vanishing, the cohomological criterion for ampleness.
Ext groups and sheaves: definition and basic properties.
Serre duality: the dualizing sheaf and duality for a projective scheme, invertibility of the dualizing sheaf for a local complete intersection, the isomorphism between the dualizing sheaf and the canonical sheaf for a nonsingular projective variety.
Schemes, cohomology and formal schemes
Schemes:
Sheaves: morphisms, the direct image functor, the inverse image functor.
Schemes: locally ringed spaces, affine schemes, \(\Proj\) of a graded ring, morphisms, the functor from varieties over an algebraically closed field \(k\) to schemes over \(k\).
First properties of schemes: integral and (locally) Noetherian schemes, morphisms locally of finite type, of finite type, or finite; open and closed subschemes; dimension; fibre products.
Separated and proper morphisms: the valuative criteria of separatedness and properness.
Sheaves of modules: constructions, locally free sheaves, sheaves of ideals, direct and inverse image, quasicoherent and coherent sheaves, the sheaf associated to a graded module, twisted sheaves, very ample invertible sheaves, globally generated sheaves.
Divisors: Weil divisors, the divisor class group, divisors on curves, Cartier divisors, the equivalence of Weil and Cartier divisors, invertible sheaves.
Projective morphisms: morphisms to projective space, ample invertible sheaves, linear systems, blowing up.
Differentials: the module of relative differentials, sheaves of differentials, nonsingular varieties, applications, local algebra.
Formal schemes: inverse limits of sheaves, formal schemes.
Cohomology of sheaves:
Derived functors.
Cohomology of sheaves: flasque sheaves, Grothendieck’s vanishing theorem.
Cohomology of a Noetherian affine scheme: Serre’s characterization of affine schemes among Noetherian schemes.
Čech cohomology.
Cohomology of projective space.
Ext groups and Ext sheaves.
Serre duality: duality for projective space over a field, the dualizing sheaf, duality for a projective scheme.
Higher direct images of sheaves.
Flat morphisms.
Varieties, schemes, cohomology and curves
Varieties: quasiprojective varieties, regular and rational maps, the definition of singularities and analytic isomorphism, abstract nonsingular curves, intersections in projective space.
Schemes: sheaves, schemes, basic definitions and properties, separated and proper morphisms and relative geometry, sheaves of modules, divisors and vector bundles (the Picard group in each of its forms), projective morphisms, differentials.
Cohomology: the formalism of derived functors, Čech cohomology, the vanishing theorem for Noetherian affine schemes, cohomology of projective space, Serre duality with Ext groups.
Curves: Riemann surfaces, Hurwitz’s theorem, local complete intersections, resolution of curve singularities, elliptic and hyperelliptic curves, group actions, Riemann–Roch with applications, degree of projective curves, the Abel–Jacobi map and Abel’s theorem.
Complex algebraic surfaces
Cohomology: the Riemann–Roch theorem, Noether’s formula, the genus formula.
Del Pezzo surfaces: definitions and blowups.
K3 surfaces: definition, cohomology, periods, polarizations, examples.
Enriques surfaces: examples and the correspondence with K3 surfaces.
ADE singularities: quotients, equations and resolutions.
The Enriques–Kodaira classification.
The Bogomolov–Miyaoka–Yau inequality.
Toric varieties
Definitions: convex polyhedral cones, geometry of convex sets, affine toric varieties, fans and toric varieties, toric varieties from polytopes.
Singularities and compactness: local properties of toric varieties, surfaces, quotient singularities, one-parameter subgroups, limit points, compactness and properness, nonsingular surfaces, resolution of singularities.
Orbits, topology and line bundles: orbits, fundamental groups and Euler characteristics, divisors, line bundles, cohomology of line bundles, the canonical class.
Cohomology of smooth toric varieties: the Stanley–Reisner relation, self-intersection of invariant divisors.
Complex algebraic geometry (minor topic)
Complex geometry (Griffiths–Harris, Ch. 0.1–0.2, 0.4–0.7, 1.1–1.4):
Complex and Kähler manifolds.
The Hodge and Lefschetz decompositions.
Holomorphic vector bundles and line bundles.
Kodaira vanishing and Kodaira embedding.
Riemann–Roch and Abel’s theorem (Miranda, Ch. V–VIII):
Divisors, embeddings, the Abel–Jacobi map.
Riemann–Roch, geometric Riemann–Roch, Abel’s theorem.
Cohomology (Griffiths–Harris, Ch. 0.3, 1.1; Miranda, Ch. IX):
De Rham, Dolbeault and Čech cohomology.
The de Rham and Dolbeault theorems.
Chern classes.
Texts: Griffiths and Harris, Principles of Algebraic Geometry; Miranda, Algebraic Curves and Riemann Surfaces.