Varieties, Morphisms & Birational Geometry

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Abstractions about algebraic varieties, morphisms, line bundles, canonical rings, blowups, moduli schemes, motivic integration, and birational invariants.

12 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Ample line bundle — A line bundle whose sufficiently high tensor power gives an embedding of a projective variety into projective space, expressing algebro-geometric positivity.
• Canonical ring — The graded ring formed from global sections of all nonnegative tensor powers of a variety's canonical bundle or canonical divisor.
• Finite morphism — A morphism of schemes that is affine and whose induced coordinate-ring algebra is finite as a module, generalizing maps with algebraically finite fibers.
• Hurwitz scheme — An algebraic moduli scheme parameterizing branched covers of a fixed target curve, commonly degree-d genus-g covers of the projective line with specified ramification data.
• Iitaka dimension — An invariant measuring asymptotic growth of sections of powers of a line bundle, equivalently the dimension of the image of its associated rational maps.
• Motivic integration — An algebraic-geometric integration theory assigning classes in a Grothendieck ring to measurable subsets or functions on arc spaces, retaining geometric information beyond numerical measure.
• Nash blowing-up — Replace a singular variety by the closure of the graph of its smooth-point tangent-space map, retaining each singular point together with the limiting tangent spaces approached nearby.
• Pseudo-canonical variety — An algebraic variety whose canonical divisor or class is pseudo-ample under the convention used, placing it in the general-type side of birational classification.
• Rosati involution — The positive involutive anti-automorphism of the rational endomorphism algebra of a polarized abelian variety obtained by taking the dual endomorphism and conjugating through the polarization.
• Ruled variety — An algebraic variety birational to a product with a projective line, so its generic points lie on a rational one-parameter ruling.
• Severi–Brauer variety — An algebraic variety over a field that becomes projective space after extending scalars to an algebraic closure, encoding a central simple algebra and its splitting.
• Toric variety — An algebraic variety containing an algebraic torus as a dense open subset whose self-action extends to the entire variety.