Algebraic Geometry skills for AI agents

Activate this skill when the user is working with the foundations of algebraic geometry: zero sets of polynomials, the Zariski topology, the dictionary between ideals and varieties, projective closures, dimension and singular points. Triggers on "affine variety," "projective variety," "Zariski topology," "Nullstellensatz," "homogenization," "projective closure," "irreducible components," "coordinate ring," "Jacobian criterion," "singular point," "twisted cubic," "smooth conic," or "algebraic geometry" foundations. Covers the ideal-variety correspondence, dimension theory, smoothness, and worked examples on the twisted cubic, conics and elliptic curves, each verified in Macaulay2, Singular or SageMath.

Activate this skill when the user is working with algebraic curves: genus, divisors, linear systems, the Riemann-Roch theorem and its consequences, elliptic curves and their group law, hyperelliptic curves, ramified covers and embeddings into projective space. Triggers on "algebraic curve," "Riemann-Roch," "genus," "divisor," "linear system," "canonical divisor," "canonical embedding," "hyperelliptic," "elliptic curve," "group law," "Weierstrass form," "Hurwitz formula," "Riemann-Hurwitz," "ramification," "degree-genus formula," "very ample," or curves in "algebraic geometry." Covers precise statements, worked Riemann-Roch computations for genus 0 through 3, the derivation of Weierstrass form and the group law, and how to check each computation in SageMath, Macaulay2 or Singular.

Activate this skill when the user has an algebraic geometry problem to attack rather than a definition to look up: a qualifying exam question, a lemma a paper needs, a conjecture to test, a proof a referee has questioned. Triggers on "how do I prove," "dimension count," "incidence correspondence," "general member," "generic," "Bertini," "semicontinuity," "local coordinates," "affine chart," "compute an example," "Gröbner check," "reduce mod p," "specialization," "degeneration," "standard tricks," "write the proof," "referee," "algebraic geometry exercise," or "algebraic geometry" problem solving. Covers the order of attack, dimension counting done rigorously, local coordinates and charts, choosing the right cohomological tool, what a computation proves and what it does not, how to write an argument that survives refereeing, and a catalogue of the tricks working geometers reach for first.

Activate this skill when the user needs the commutative algebra that algebraic geometry actually runs on, with every theorem tied to its geometric meaning. Triggers on "Noetherian ring," "localization," "local ring," "integral extension," "integral closure," "normalization," "Noether normalization," "Krull dimension," "primary decomposition," "associated primes," "embedded component," "regular local ring," "Cohen-Macaulay," "Hilbert polynomial," "Hilbert series," "depth," or requests to translate between ring theory and geometry. Covers the theorems, what each one says about a variety, and how to compute each invariant in Macaulay2, Singular and SageMath.

Activate this skill when the user wants to compute with polynomial ideals: test membership, eliminate variables, solve polynomial systems, implicitize a parametrization, or understand why a computation will not finish. Triggers on "Gröbner basis," "Groebner basis," "monomial order," "lex order," "grevlex," "Buchberger algorithm," "S-polynomial," "division algorithm," "elimination ideal," "implicitization," "solving polynomial systems," "zero-dimensional ideal," "Macaulay2," "Singular," "SageMath," or computational "algebraic geometry." Covers the theory precisely, real sessions in Macaulay2, Singular and Sage with actual syntax, and the complexity facts that decide which computations are feasible.

Activate this skill when the user needs to count intersections of subvarieties with the correct multiplicities, work with divisors and intersection numbers on surfaces, or reason about blow-ups and exceptional curves. Triggers on "Bézout's theorem," "Bezout," "intersection multiplicity," "intersection number," "Chow group," "Chow ring," "self-intersection," "(-1)-curve," "exceptional divisor," "blow-up," "adjunction formula," "Hodge index," "Riemann-Roch for surfaces," "27 lines," "cubic surface," or intersection-theoretic "algebraic geometry." Covers the local definition of multiplicity, Bézout in the plane and in P^n, the Chow ring of projective space and of products, the intersection pairing on surfaces with blow-ups and adjunction, and worked surface examples checked in Singular, Macaulay2 and Sage.

Activate this skill when the user is learning or using the language of schemes: Spec and Proj, structure sheaves, generic points, nilpotents, fibre products, and the properties of morphisms that carry the geometry. Triggers on "scheme," "Spec," "Proj," "structure sheaf," "generic point," "nilpotent," "fibre product," "closed immersion," "open immersion," "separated," "proper morphism," "finite type," "finite morphism," "flat family," "flat limit," "valuative criterion," or "why schemes" in algebraic geometry. Covers the definitions with their reasons, the standard morphism properties and how to check them, flatness as continuity of fibres, and concrete computations of fibres and flat limits in Macaulay2 and Sage.

Activate this skill when the user needs to work with sheaves on varieties and schemes and to compute or use sheaf cohomology. Triggers on "sheaf," "quasi-coherent," "coherent sheaf," "line bundle," "invertible sheaf," "divisor," "Picard group," "Cech cohomology," "Čech cohomology," "sheaf cohomology," "H^1," "Serre duality," "canonical bundle," "vanishing theorem," "Kodaira vanishing," "Serre vanishing," "Castelnuovo-Mumford regularity," "long exact sequence," "ideal sheaf sequence," "Euler characteristic," "Hilbert polynomial," or cohomological "algebraic geometry." Covers the definitions, the divisor-line bundle dictionary, Čech computations done by hand on projective space, the statement and use of Serre duality, the vanishing theorems that actually get used, Euler characteristic bookkeeping, and how to check every number in Macaulay2, Singular or SageMath.

Activate this skill when the user is building or analysing toric varieties: turning cones, fans and lattice polytopes into varieties, reading smoothness, completeness and projectivity off the fan, computing torus-invariant divisors, their polytopes and sections, or resolving toric singularities by subdivision. Triggers on "toric variety," "fan," "rational polyhedral cone," "lattice polytope," "normal fan," "orbit-cone correspondence," "torus-invariant divisor," "Cox ring," "Hirzebruch surface," "weighted projective space," "Demazure vanishing," "Ehrhart polynomial," "Hilbert basis," "toric ideal," "reflexive polytope," "toric Fano," or combinatorial "algebraic geometry." Covers the cone-fan-polytope dictionary with precise statements, the criteria that decide geometry from combinatorics, divisors and cohomology through lattice points, and working sessions in SageMath, Macaulay2 and Singular.