Alphabeta MathZariski Tangent Spaces, Regular Points, Smoothness, and Bertini
About this page
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 37 results · all verified · 23 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 14 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Dimension Constructible Images and Dimensions of Fibres
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Integral Extensions and Going Up
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Functors
- Ext and Balanced Resolutions
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
- Topology of ℝ
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Presheaves Sheaves Stalks and Sheafification
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Definition
The intrinsic cotangent space
Definition
The intrinsic Zariski tangent space
Lemma
Cotangent spaces commute with localization at a rational point
Lemma
Tangent vectors at rational points are dual-number points
Lemma
Square-zero vector extensions encode tangent vectors with coefficients
Definition
Equation rows and coordinate columns in an affine Jacobian
Theorem
The Jacobian kernel computes the tangent space
Lemma
Differentials, open restriction, and the chain rule
Lemma
Tangent spaces of products over a field
Definition
Regular points of locally Noetherian schemes
Lemma
Local dimension for a reducible classical algebraic set
Theorem
Tangent dimension bounds local dimension
Definition
Regular and singular loci
Theorem
Jacobian rank detects regularity at closed points
Corollary
The gradient test for a reduced hypersurface
Lemma
A regular point lies on one irreducible component
Theorem
Openness of the regular locus over a perfect field
Lemma
A dense hypersurface chart with a nonzero partial derivative
Theorem
Dense regular loci on every component
Corollary
Minimal tangent dimension and homogeneous regularity
Definition
Smoothness over a field by geometric regularity
Theorem
Regular equals smooth over a perfect field
Theorem
Purely inseparable field algebras separate regularity from smoothness
Definition
The scheme-theoretic tangent cone at a point
Lemma
All initial forms define the tangent cone
Lemma
The scheme-theoretic linear span of the tangent cone
Definition
Multiplicity of a hypersurface equation at a rational point
Lemma
Multiplicity one is the smooth hypersurface test
Lemma
Products preserve smoothness
Definition
Smooth morphisms via local standard smooth presentations
Lemma
The submersion criterion between smooth varieties
Lemma
A transverse hyperplane slice is smooth at the chosen point
Lemma
A tangent direction is realized by a local smooth curve
Lemma
A dominant map has a surjective differential on a dense source open
Corollary
Generic smoothness on the source
Lemma
Critical loci have small images in characteristic zero
Theorem
Generic smoothness over a dense target open
Lemma
A section of an invertible sheaf has a canonical zero subscheme
Definition
Linear systems, base loci, and general members
Lemma
The universal member away from the base locus
Theorem
Bertini smoothness away from the base locus
Corollary
General hypersurfaces give smooth complete intersections
Remark
Conventions and hypotheses carried by this pair
Sources
- Milne, Algebraic Geometry, §4f, Proposition 4.29 and §4.30
- Stacks Project, Varieties, Section 33.16, tag 0B28
- J. S. Milne, Algebraic Geometry, v6.10, §4f, Proposition 4.29 and item 4.30
- The Stacks Project, Varieties, Section 33.16, tangent spaces (tag 0B28)
- J. S. Milne, Algebraic Geometry, v6.10, Ch. 1 §b Lemma 1.15; Ch. 4 §f item 4.30(d) and §g
- J. S. Milne, Algebraic Geometry, v6.10, §4f, Definitions 4.25 and 4.28, Propositions 4.27 and 4.29, and item 4.30
- J. S. Milne, Algebraic Geometry Chapter 10 supplement, §f, item 10.60
- J. S. Milne, Algebraic Geometry, v6.10, §4f, items 4.27–4.30, and Exercise 4-10
- Milne, Algebraic Geometry, §4d, Definition 4.22
- J. S. Milne, Algebraic Geometry, v6.10, §4d Definition 4.22 and §4f Proposition 4.26
- J. S. Milne, Algebraic Geometry, v6.10, §4e Lemma 4.24 and §4f item 4.31
- Donu Arapura, Algebraic Geometry, Lemma 5.1.5
- J. S. Milne, Algebraic Geometry, v6.10, Exercise 4-4 and its solution
- J. S. Milne, Algebraic Geometry, v6.10, §4i, Theorem 4.44 and Corollary 4.45
- J. S. Milne, Algebraic Geometry, Ch. 10 supplement, §f, 10.58–10.59
- Milne, Algebraic Geometry, §3c notes 3.13–3.14 and §4i, proof of Corollary 4.45
- Milne, Algebraic Geometry Chapter 10 supplement, §§10.54–10.56
- J. S. Milne, Algebraic Geometry, v6.10, §4i, Theorem 4.44, with its cited arguments in 4.36 and 3.45
- J. S. Milne, Algebraic Geometry, v6.10, §4h, Definition 4.35; §4i, Corollary 4.45
- J. S. Milne, Algebraic Geometry v6.10, §4d, Definition 4.23 and Jacobian-rank discussion (printed pp. 87–88; PDF pp. 86–87), and §4i, Corollary 4.45 (printed p. 97)
- J. S. Milne, Algebraic Geometry Chapter 10 supplement, §f, items 10.58 and 10.60–10.64
- The Stacks Project, Algebra Lemma 10.140.4 (tag 00TU), separable-residue cotangent injection
- The Stacks Project, Algebra Lemma 10.140.5 (tag 00TV), regularity and smoothness with separable residue field
- J. S. Milne, Algebraic Geometry v6.10, §4a Definition 2.1 and Examples 4.5–4.6; §4d Definitions 4.22–4.23 and the hypersurface Jacobian criterion; §4h Definition 4.35
- J. S. Milne, Algebraic Geometry, v6.10, §4i, proof of Corollary 4.45, printed p. 97
- J. S. Milne, Algebraic Geometry, v6.10, §4h, Theorem 4.37 (printed p. 95; PDF p. 94)
- The Stacks Project, Varieties Lemma 33.25.8 (tag 0B8X), regular locus equals smooth locus over a perfect field
- Milne, Algebraic Geometry, §3k Propositions 3.36–3.38 and §4h proof of Theorem 4.37
- J. S. Milne, Algebraic Geometry, v6.10, §4h, Theorem 4.37 (printed p. 95, PDF p. 94)
- J. S. Milne, Algebraic Geometry, v6.10, §4h, Corollaries 4.38-4.40 (printed p. 95)
- Donu Arapura, Notes on Basic Algebraic Geometry, §5.2, Corollary 5.2.4 (homogeneous regularity)
- The Stacks Project, Varieties, Definition 33.12.1 and Lemmas 33.12.3 and 33.12.6 (tag 038S)
- The Stacks Project, Varieties Lemma 33.12.3 (tag 038V), affine chart criterion for geometric regularity
- The Stacks Project, Varieties Lemma 33.12.6 (tag 038X), geometric regularity and smoothness at a point
- The Stacks Project, Algebra Lemma 10.166.1 (tag 0381), finite purely inseparable field test
- J. S. Milne, Algebraic Geometry, Chapter 10 supplement, §f, item 10.64 (printed p. 18; PDF page 18)
- The Stacks Project, Varieties, Example 33.12.7 (tag 038S): $\operatorname{Spec}(k[x]/(x^p-t))$ is a regular variety that is not geometrically reduced
- J. S. Milne, Algebraic Geometry, v6.10, §4g, tangent cones and Proposition 4.34
- J. S. Milne, Algebraic Geometry Chapter 10 supplement, Definitions 10.69–10.71
- The Stacks Project, Section 27.7, Cones, Definitions 27.7.1–27.7.2 (tag 062P)
- J. S. Milne, Algebraic Geometry, v6.10, §4g, warning before Proposition 4.34 and Proposition 4.34
- J. S. Milne, Algebraic Geometry, Ch. 10 supplement, §f, Definitions 10.69–10.70
- J. S. Milne, Algebraic Geometry, v6.10, §4g, Proposition 4.34
- J. S. Milne, Algebraic Geometry Chapter 10 supplement, §f, Definitions 10.69–10.71
- J. S. Milne, Algebraic Geometry, v6.10, §4b, Definition 4.9 and following multiplicity paragraph
- J. S. Milne, Algebraic Geometry v6.10, §4b, Definition 4.9 and following multiplicity paragraph
- J. S. Milne, Algebraic Geometry, Chapter 10 supplement (AG10), §f, items 10.58 and 10.64
- Ravi Vakil, Foundations of Algebraic Geometry, Classes 51–52, §2.8
- The Stacks Project, Morphisms of Schemes, Lemmas 29.35.4–5 and 29.35.11 (tag 01V4)
- J. S. Milne, Algebraic Geometry v6.10, §5j, Proposition 5.35
- Stacks Project, Algebra, Lemma 10.137.16 (tag 00TF)
- Ravi Vakil, Foundations of Algebraic Geometry, Classes 51–52, §§1.2, 1.9, 2.12
- Ravi Vakil, Foundations of Algebraic Geometry, Classes 51–52, §2.2, Trickier Exercise
- The Stacks Project, Algebra Lemma 10.128.2 (tag 07DY), regular parameters mapping to a regular sequence imply flatness
- J. S. Milne, Algebraic Geometry, v6.10, Exercise 4-2 (printed pp. 98-99) with its solution (printed p. 222)
- J. S. Milne, Algebraic Geometry, v6.10, Exercise 4-3 (printed p. 98) with its solution (printed p. 222)
- Ravi Vakil, MATH 216 (2005-06), Classes 51-52, §3.1, Proposition 3.1 (generic smoothness in the source) and its proof
- Ravi Vakil, MATH 216 (2005-06), Classes 51-52, §3.1, Proposition 3.1 (generic smoothness in the source) with proof
- Ravi Vakil, MATH 216 (2005-06), Classes 51-52, §3.4 (Lemma on the dimension of the critical image) with proof
- Ravi Vakil, MATH 216 (2005-06), Classes 51-52, §3.3, Theorem 3.3 (generic smoothness in the target) with proof
- Donu Arapura, Notes on Basic Algebraic Geometry, Theorem 5.4.2 (Bertini-Sard), printed p. 34
- Vakil, Foundations of Algebraic Geometry Classes 51–52, §3.9 Corollary 3.9, p. 10
- Vakil, Foundations of Algebraic Geometry Classes 51–52, §3.9 Corollary 3.9, printed p. 10
- Arapura, Notes on Basic Algebraic Geometry, §5.4 hyperplane parameterization and Theorem 5.4.5, printed pp. 38–39
- Donu Arapura, Notes on Basic Algebraic Geometry, proof of Theorem 5.4.5, printed p. 39
- Ravi Vakil, Foundations of Algebraic Geometry, Classes 51–52, §3.9, printed pp. 9–11
- Ravi Vakil, MATH 216 (2005-06), Classes 51-52, §3.9 Corollary and §3.11 (Bertini), printed pp. 10-11
- Donu Arapura, Notes on Basic Algebraic Geometry, §5.4 Theorem 5.4.5 and its proof, printed pp. 38-39
- Donu Arapura, Notes on Basic Algebraic Geometry, §5.4, Theorem 5.4.5 and the discussion preceding it, printed pp. 38–39; general hypersurface statement on p. 39
- Ravi Vakil, MATH 216 (2005-06), Classes 51–52, §3.9 Corollary and §3.10–3.11, printed pp. 10–11
- Robin Hartshorne, Algebraic Geometry, Chapter II, Theorem 8.18 and Chapter III, Corollary 10.9
- J. S. Milne, Algebraic Geometry, Ch. 4 §§d–i (Jacobians, tangent cones, regularity)
- Donu Arapura, Notes on Basic Algebraic Geometry, §5.4