# Oral Exam Syllabus



The presentation is based on the following paper: 

- Engel, [Looijenga's Conjecture via Integral-affine Geometry](https://arxiv.org/abs/1409.7676)


The scope of topics for the examination portion is as follows:



## Varieties

At the level of [Hart Ch.1]


- Affine varieties
- Projective varieties
- Morphisms
- Rational maps
- Nonsingular varieties
- Nonsingular curves



## Schemes

At the level of [Hart Ch.2]


- Sheaves: 

   - Presheaves, sheaves, sheafification, morphisms of (pre)sheaves, pullback and pushforward
   - Skyscraper sheaves, constant sheaves, sheaf hom, sheaf image/kernel/cokernel
- Schemes: 

   - Ringed spaces and their morphisms, the structure sheaf
   - Generic points and closed points, residue fields
   - Definitions of affine/projective/general schemes, the Proj construction
   - Morphisms of finite type, finite morphisms, open/closed immersions
   - Affine and projective morphisms
   - Dimension, Krull dimension
   - Fiber products
- First properties of schemes: 

   - Reduced, irreducible, integral, Noetherian schemes
   - Morphisms of finite type, finite morphisms, 
   - Open/closed immersions, dimension, fiber products
- Separated and proper morphisms: 

   - Definition of separated and proper morphisms, 
   - Valuative criteria for being separated, universally closed, proper.
   - Projective morphisms, 
   - Reduced scheme structure of a closed subset, 
   - Scheme-theoretic image, 
- Coherent and quasi-coherent sheaves: 

   - Definition of $\mathcal{O}_X$-modules, 
   - Quasi-coherent and coherent sheaves/$\mathcal O_X$-modules, 
   - Sheaves of ideals, direct image and inverse image
   - The $\mathcal{O}_X$-module associated to a module over a ring.
- Divisors:

   - Invertible sheaves and vector bundles
   - Cartier divisors and their associated invertible sheaves; 
   - Multiplicity and support of Cartier divisors; 
   - Cartier divisors of rational sections of invertible sheaves; 
   - Weil divisors and relation with Cartier (equivalence for locally factorial schemes) 
   - The Picard group and class group
   - Degree of a divisor
- Projective morphisms: 

   - Ample and very ample sheaves, 
   - Blowups
- Differentials:

   - Derivations, module of relative (algebraic) differential forms
   - The tangent sheaf and canonical sheaf
   - Dualizing sheaf and Serre duality, relation to the canonical sheaf
   - The normal and conormal sheaves, 
   - Local complete intersections; 
- Misc

     - Normal schemes and normalization
     - The adjunction formula, crepant resolutions
     - Geometric and arithmetic genus
     - Riemann-Roch, Riemann-Hurwitz


   

## Cohomology

At the level of [Hart Ch.3]

1. Derived functors: 
   Abelian categories, chain complexes, derived functors, $\delta$-functors
2. Cohomology of sheaves: 
   The category of sheaves of $\mathcal{O}_X$-modules has enough injectives, Serre/Grothendieck vanishing
3. Cohomology of Noetherian affine schemes: 
   Characterization of noetherian affine schemes by vanishing of sheaf cohomology for quasicoherent sheaves
4. Čech cohomology:
   Definition, isomorphism with sheaf cohomology for a Noetherian separated scheme
5. Cohomology of the structure sheaf for $\mathbf{P}^n$
6. Higher direct images and pushforwards of sheaves
7. Flat morphisms, flat families, smooth morphisms
8. Flasque and fine sheaves
9. Zariski's main theorem




## Toric Varieties



At the level of [Fulon Ch.1,2,3,4]


1. Definitions: 
   - The dual character/cocharacter lattices $M$ and $N$, 
   - Strongly convex rational polyhedral cones and their dual cones
   - Combinatorics and basic convex geometry of cones and fans
   - Constructing affine toric varieties from semigroup algebras, and projective toric varieties from gluing affines
   - Basic examples of fans: $(\mathbf{C}^\times)^n, \mathbf{C}^n, \mathbf{P}^n,$ the $n$th Hirzebruch surface $\mathbf{F}_n$, the quadric cone (cone over a conic), the rational cuspidal curve, weighted projective spaces, and products and blowups thereof
   - Toric varieties from polytopes, polar duals, reflexive polytopes
   - The orbit-cone correspondence, orbit closures
   - Toric morphisms
2. Singularities and compactness: 
   - Checking smoothness, properness/completeness
   - Sufficient conditions for orbifold/quotient singularities
   - Torus-invariant Weil divisors associated to rays
   - Cartier divisors associated to integral piecewise-linear support functions
   - Blowups and resolution of singularities
   - Canonical divisors of toric varieties, Fano toric varieties
   - Classification of complete nonsingular toric surfaces
   - Classification of smooth del Pezzo surfaces
3. Orbits, topology, and line bundles: 

   - Characterization of (very) ample line bundles
   - Global sections of line bundles
   - Computing fundamental groups and Euler characteristics. 
   - $T$-Cartier and $T$-Weil divisors and computing their intersection numbers
   - Computing divisor class groups and Picard groups



# References