Basics
- For \(V\) a affine variety, what is \(k[V]\)? What is \(k(V)\)?
- What is a Noetherian space?
- What is \({\mathcal{O}}_X\)?
- How is \({\mathcal{O}}_X\) defined for \(X \subseteq {\mathbb{A}}^n\)?
- What are the Zariski closed subsets of \({\mathbb{A}}^n\)?
- What are the maximal ideals of \({\mathcal{O}}_X(X)\)?
- What is a distinguished open set?
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Types of varieties:
- What is an affine variety? A projective variety?
- What is an irreducible subset of an affine variety?
- What is a proper variety?
- What is a complete variety?
- What is a normal affine variety?
- What is a Fano variety?
- What is a Calabi-Yau variety?
- What is a toric variety?
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Properties of varieties
- What is the dimension of an affine variety?
- What is the degree of a variety?
- What is a smooth point of a variety? A singular point?
- What is a separated variety?
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Constructions:
- What is the Unsorted/normalization of an affine variety?
- What is the projective closure of an affine variety
- Given a projective variety \(X\), what is its affine cone?
- What is the Hilbert function of a variety? The Hilbert polynomial?
- What is the rational function field \(\kappa_X\) of an affine variety?
- What does it mean for elements of \(\kappa_X\) to be algebraically independent?
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What is a hyperplane section?
- What is the hyperplane at \(\infty\) of \({\mathbb{P}}^n\) in coordinates?
- What is the degree of a \(k{\hbox{-}}\)dimensional subvariety of \({\mathbb{P}}^n\)?
- What is the Néron-Severi group?
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Special bundles:
- What is the canonical bundle? The anticanonical? Pluricanonical bundles?
- What is the tautological bundle?
- What is the Serre twisting line bundle?
- What does it mean for a bundle to be positive?
- What is the Segre embedding?
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Singularities:
- What is the Zariski tangent space?
- What is a simple/smooth/regular/non-singular point of an affine variety?
- What is a cuspidal singularity?
- What is a nodal singularity?
- What is a normal crossing singularity?
- What is a local complete intersection?
- What is an embedding of projective varieties?
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What is the blowup of a point? The blowdown?
- What is the proper transform? The strict transform?
- What is the exceptional curve/divisor?
- What is a Chow variety?
- What is the Grassmannian?
- What is a tangent cone?
- What is a dual variety?
- What is a hypersurface?
- What is a very general hypersurface?
- What does it mean to be unirational?
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What does it mean to be ruled?
- What does it mean to be uniruled?
- What is a scroll?
- What is the gonality of a curve?
- What is a linear series?
- What is specialization?
Morphisms
- What is a regular function?
- What is a Unsorted/birational?
- What is a dominant morphism between varieties?
- What is a birational morphism?
- What is a rational function on a variety?
- What is a finite morphism?
- What is a quasi-finite morphism?
- What is the degree of a morphism?
- What is a closed morphism?
- What is a rational map of affine varieties?
- What is the indeterminacy set of a rational map?