Basics
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For \(V\) a projective variety, what is \(k[V]\)? What is \(k(V)\)
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What is a Noetherian space?
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What is an affine variety? A projective variety?
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What is the projective closure of an affine variety
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What is the dimension of an affine variety?
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Given a projective variety \(X\), what is its affine cone?
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What are the Zariski closed subsets of \({\mathbb{A}}^n\)?
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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 the normalization of an affine variety?
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How is \({\mathcal{O}}_X\) defined for \(X \subseteq {\mathbb{A}}^n\)?
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What are the maximal ideals of \({\mathcal{O}}_X(X)\)?
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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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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 is the Segre embedding?
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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 an embedding of projective varieties?
Morphisms
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What is a birational morphism?
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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 dominant map between affine varieties?
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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?
Geometry
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What is a Fano variety?
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What is a del Pezzo surface?
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What is a ruled variety?
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What is a uniruled variety?
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What is the Kodaira dimension?
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What is an elliptic curve?
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What is a hyperelliptic curve?
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What is…
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An Enriques surface?
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A K3 surface?
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A hyperelliptic surface?
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An abelian surface?
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An elliptic surface?
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What is the blowup of a point? The blowdown?
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What is the proper/strict transform?
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What is the exceptional curve/divisor?
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What is the genus of a curve? The geometric genus? The arithmetic genus?
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What is the Todd genus?
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What is the Jacobian of a curve?
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What is a minimal model of a smooth projective variety?
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What is an elliptic fibration?