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