Alphabeta MathAlgebraic Geometry
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
5 pagesThis page sets up the classical affine dictionary over an algebraically closed field.
7 definitions, 7 lemmas, 6 theorems, 1 corollary, 1 remarkExamples & counterexamples →Standard smoothness is characterised here by flatness together with geometrically regular fibres, and the criterion is reached through Kähler differentials: the universal…
4 definitions, 14 lemmas, 8 theorems, 1 exampleExamples & counterexamples →This page develops the classical affine dictionary over an algebraically closed field.
18 definitions, 17 lemmas, 14 theorems, 1 exampleExamples & counterexamples →This page develops groups of multiplicative type over a field and their arithmetic classification.
4 definitions, 6 lemmas, 1 theorem, 1 corollaryExamples & counterexamples →This page develops rational mixed Hodge structures as three opposed filtrations and proves the two structural results the rest of the mixed-Hodge track uses: the Deligne splitting and the strictness of morphisms.
2 definitions, 3 lemmasExamples & counterexamples →
Level 1
4 pagesThis page sets up the analytic topology of a complex algebraic variety, the finite smooth algebraic stratifications with the frontier condition that organize it, and the…
3 definitions, 3 lemmasExamples & counterexamples →This page develops the classical complex theory of algebraic group actions on affine algebraic sets and the finite-dimensional linear models of those actions.
1 definition, 3 lemmas, 1 proposition, 2 theoremsExamples & counterexamples →This page upgrades the affine algebraic-set dictionary from objects to morphisms.
12 definitions, 4 lemmas, 6 theorems, 1 corollaryExamples & counterexamples →This page fixes classical projective coordinates over an algebraically closed field, then develops homogeneous equations, affine charts, projective closure and saturation, regular functions, coordinate morphisms, and affine cones.
11 definitions, 9 lemmas, 4 theorems, 1 remarkExamples & counterexamples →
Level 2
1 pageOver a fixed algebraically closed field k, this page constructs the classical affine and projective products used here, then gives coordinate models for the Segre, Veronese, and Plucker maps.
7 definitions, 8 lemmas, 5 theorems, 4 corollaries, 1 remarkExamples & counterexamples →
Level 3
1 pageDimension is measured by chains of nonempty irreducible closed subsets, with the empty value -∞.
8 definitions, 18 lemmas, 11 theorems, 8 corollaries, 1 remarkExamples & counterexamples →
Level 4
1 pageThe page builds the tangent space from the cotangent space. For a point of a finite-type k-scheme the intrinsic cotangent space is m x/m x² over the residue field κ(x), and…
10 definitions, 19 lemmas, 9 theorems, 4 corollaries, 1 remarkExamples & counterexamples →
Level 5
7 pagesThis page builds the finite incidence and cochain models that turn constructible sheaves on a finite simplicial complex into finite diagrams, and the locally split Čech…
6 lemmasExamples & counterexamples →This page develops Grothendieck's finite étale descent and the étale fundamental group of a connected scheme, together with the smooth proper specialization theorem that…
1 definition, 15 lemmas, 8 theoremsExamples & counterexamples →This page develops the numerical intersection product on an integral regular projective surface over a field.
2 definitions, 4 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →This page develops the structure of nonaffine algebraic groups over a field.
1 definition, 49 lemmas, 2 propositions, 17 theoremsExamples & counterexamples →The page develops normality, normalization, finite morphisms and the classical form of Zariski's Main Theorem.
5 definitions, 12 lemmas, 9 theorems, 4 corollaries, 1 remarkExamples & counterexamples →The page develops the classical local intersection calculus of plane projective curves over an algebraically closed field, assuming the Axiom of Choice for the cited classical and length results, and proves Bézout's theorem.
8 definitions, 13 lemmas, 4 theorems, 5 corollaries, 1 remarkExamples & counterexamples →This page develops projective-space cohomology and its residue pairing, then constructs the smooth-projective trace and Serre pairing through regular-immersion and Koszul calculations.
4 definitions, 26 lemmas, 9 theoremsExamples & counterexamples →
Level 6
8 pagesThis page develops the theory of abelian varieties over a field together with their arithmetic models over a discrete valuation ring.
8 definitions, 50 lemmas, 7 theorems, 2 corollariesExamples & counterexamples →This page develops Chow groups and intersection theory from the cycle level up, then proves Grothendieck-Riemann-Roch for projective morphisms.
10 definitions, 22 lemmas, 5 theorems, 1 remarkExamples & counterexamples →This page develops duality for coherent sheaves on a projective scheme X over a field k that is Cohen–Macaulay and pure of dimension d, with no smoothness assumption: X may…
1 definition, 7 lemmas, 1 theorem, 2 remarksExamples & counterexamples →- Complex Algebraic Stratified Topology15 results
This page develops complex algebraic stratified topology: finite smooth complex algebraic Whitney stratifications of reduced quasi-projective varieties and of a single…
1 definition, 11 lemmas, 3 theoremsExamples & counterexamples → This page proves embedded resolution of a reduced curve on an arbitrary Noetherian regular surface by finitely many point blowups, assuming that each irreducible component…
2 definitions, 7 lemmas, 3 theoremsExamples & counterexamples →This page develops the affine invariant theory of a complex reductive affine algebraic group and the geometric quotient of its stable locus.
3 definitions, 7 lemmas, 3 theoremsExamples & counterexamples →This page develops the Riemann-Roch theorem and the Hodge index theorem for an integral smooth projective surface over a field.
2 definitions, 5 lemmas, 3 theorems, 1 corollary, 1 remarkExamples & counterexamples →This page develops the structure theory of unipotent and solvable affine algebraic groups over a field and the conjugacy theorems for Borel subgroups and maximal tori over an algebraically closed field.
8 definitions, 23 lemmas, 5 propositions, 10 theoremsExamples & counterexamples →
Level 7
5 pagesThis page develops the perverse-sheaf geometry of affine varieties and of universal hyperplane families.
7 lemmasExamples & counterexamples →This page develops the birational geometry of surfaces: exceptional curves and their contractions, the negativity of curves contracted by a birational morphism, factorization…
3 definitions, 77 lemmas, 5 theorems, 1 remarkExamples & counterexamples →This page develops the characteristic-zero resolution of singularities after Hironaka in the form given by Włodarczyk's marked-ideal algorithm, over Conventions for the resolution development.
11 definitions, 38 lemmas, 1 proposition, 4 theorems, 1 remarkExamples & counterexamples →This page constructs the projective GIT quotient of a projective variety by a reductive group with respect to an ample linearized invertible sheaf, and it proves which…
4 definitions, 9 lemmas, 3 theoremsExamples & counterexamples →This page develops the structure theory of split reductive groups over a field k: maximal tori, root data, root groups, the Weyl group, the Bruhat decomposition and the theory of parabolic subgroups with their Levi decompositions.
7 definitions, 16 lemmas, 13 theoremsExamples & counterexamples →
Level 8
2 pagesThis page develops the highest-weight theory of the rational representations of a split reductive group over an arbitrary field.
5 definitions, 21 lemmas, 2 propositions, 6 theorems, 1 remarkExamples & counterexamples →This page constructs the logarithmic de Rham complex of a smooth quasiprojective complex variety and reads off the weight filtration on its open cohomology.
2 lemmas, 1 theoremExamples & counterexamples →
Level 9
2 pagesThis page assembles the geometric fixed-part and semisimplicity package for the rational Gauss-Manin local systems of smooth projective families.
1 definition, 4 lemmas, 2 theoremsExamples & counterexamples →This page constructs the proper Hodge-theoretic input for the mixed Hodge theory of singular proper varieties, and it proves the restriction-image statement used in the absolute Hodge package on this frontier.
2 definitions, 2 lemmas, 3 theoremsExamples & counterexamples →
Level 10
1 pageThis page assembles the compact-source relative hard Lefschetz, decomposition and semisimplicity theorem for a projective morphism whose source is smooth projective, together with the two local mechanisms the induction needs.
3 lemmas, 2 theoremsExamples & counterexamples →