Alphabeta MathZariski Tangent Spaces, Regular Points, Smoothness, and Bertini — Examples
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✓ 17 results · all verified · 12 also independently AI-judged
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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
- Projective Algebraic Sets Projective Morphisms and Cones
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
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
Example
The tangent space of the parabola at a general point and the local parameter
Example
A node has two distinct tangent directions
Example
The cusp retains a doubled tangent line
Example
A nondegenerate projective quadric
Counterexample
The equation must define the intended scheme
Counterexample
A regular point that is not smooth: a purely inseparable thickening
Counterexample
Frobenius linear systems have nonreduced general members
Example
The product of two parabolas: a block Jacobian and the direct-sum formula
Example
A projective cone with a smooth conic base
Counterexample
A cusp family defeats the missing source hypothesis
Example
The rank-one 2 by 2 determinantal cone
Example
Dual numbers compute the tangent spaces of the general and special linear groups
Example
Three axes are not three coplanar lines
Example
Different singularities can share a tangent cone
Example
Classical bilinear-form equations linearize to matrix spaces
Example
An irreducible curve can have arbitrarily large tangent dimension
Counterexample
A nonradical ideal need not enlarge every tangent space
Sources
- J. S. Milne, Algebraic Geometry, v6.10, §4a Definition 4.1 and equation (20) (printed pp. 81–82); §4f item 4.30(d) (printed p. 92)
- Donu Arapura, Notes on Basic Algebraic Geometry, Exercise 5.1.1 and Theorem 5.1.2 (printed pp. 34–35); Lemma 5.2.2 (printed p. 35)
- J. S. Milne, Algebraic Geometry, v6.10, Example 4.4 (printed p. 82) and Example 4.10 (printed p. 84)
- J. S. Milne, Algebraic Geometry, v6.10, Example 4.3 (printed p. 82) and Example 4.12 (printed p. 84)
- J. S. Milne, Algebraic Geometry Chapter 10 supplement, Definitions 10.67 and Example 10.68 (printed p. 18)
- J. S. Milne, Algebraic Geometry, v6.10, §4a (nonsingularity via the partial derivatives), Exercise 6-1 (printed p. 159)
- Donu Arapura, Notes on Basic Algebraic Geometry, §5.2 (nonsingularity, printed p. 35)
- J. S. Milne, Algebraic Geometry, v6.10, Example 4.2 (printed p. 82) and Exercise 4-9 (printed p. 99)
- The Stacks Project, Varieties, Example 33.12.7 (tag 038S), first example
- R. Vakil, Foundations of Algebraic Geometry, Classes 51–52, §3.11 final warning (printed p. 11)
- D. Arapura, Notes on Basic Algebraic Geometry, §5.4.3 (printed p. 39)
- J. S. Milne, Algebraic Geometry, v6.10, Exercise 4-4 (printed p. 98) and its solution (printed p. 222)
- J. S. Milne, Algebraic Geometry, v6.10, Exercise 4-8 (printed p. 99; solution printed p. 223) and Exercise 6-1 (printed p. 159)
- R. Vakil, Foundations of Algebraic Geometry (Math 216, 2005-06), Classes 51-52, §3.1 (source-open generic smoothness) and §3.3 (target-open statement with a smooth source)
- D. Arapura, Notes on Basic Algebraic Geometry, §5.4 around Theorem 5.4.2 (generic smoothness over a target open)
- Donu Arapura, Notes on Basic Algebraic Geometry, Example 5.1.3 and §5.2 (printed p. 35)
- J. S. Milne, Algebraic Geometry, v6.10, §4j Examples 4.46–4.47 (printed pp. 97–98): a matrix I+εA has inverse I−εA in M_n(k[ε]) and so lies in GL_n(k[ε]), whence T_e(GL_n)={I+εA:A∈M_n}≅M_n(k); and det(I+εA)=1+ε trace(A), whence T_e(SL_n)={I+εA:trace(A)=0}≅{A∈M_n(k):trace(A)=0}
- Donu Arapura, Notes on Basic Algebraic Geometry, Exercise 5.1.4 (printed p. 35): show that det(I+Bε)=1+trace(B)ε, and conclude that T_I Sl_n(k)={B∈Mat_{n×n}(k):trace(B)=0}
- J. S. Milne, Algebraic Geometry, v6.10, Example 4.43 (printed pp. 96–97)
- J. S. Milne, Algebraic Geometry, v6.10, Examples 4.13-4.17 (printed pp. 84-85)
- J. S. Milne, Algebraic Geometry v6.10, §4j Example 4.48, Exercise 4-6, and official solution 4-6
- J. S. Milne, Algebraic Geometry, v6.10, Exercise 4-5 (printed p. 99) with its solution (printed p. 223)
- J. S. Milne, Algebraic Geometry, v6.10, Exercise 4-9 (printed p.99) and its solution (printed p.223)