Algebraic Geometry

Introduction

Algebraic geometry is the study of the solution sets of systems of polynomial equations. The subject is generated by a single observation: the geometric properties of such a set $X$ are read from the algebra of the polynomial functions on it, and conversely the algebra of a finitely generated ring over a field is a geometric object. The translation is made exact by the Nullstellensatz, which converts the correspondence between closed subsets and ideals into a bijection between the closed subsets of affine space over an algebraically closed field and the radical ideals of a polynomial ring, and by the contravariant equivalence between the polynomial maps of affine varieties and the algebra homomorphisms of their coordinate rings. This article develops the classical theory of affine and projective varieties — the Zariski topology and its strange properties, the Nullstellensatz and its consequences, morphisms, the function field and rational maps, dimension, smoothness and singularities — and states the dictionary with commutative algebra that the whole subject rests on. The scheme-theoretic reformulation, in which the algebraically closed field, the reducedness and the embedding into affine space are all dropped, is developed, written in this same batch, and the theory of sheaves on varieties and schemes in Sheaves in Algebraic Geometry and also written in this same batch.

Two boundaries are stated at once, because the corpus distributes this material across Parts. The arithmetic theory of curves — the divisors, the genus, the group law on an elliptic curve, the Riemann–Roch theorem for curves and its number-theoretic applications — belongs to Part I's Algebraic Curves, Elliptic Curves and The Riemann–Roch Theorem for Curves; the scheme-theoretic side of the same theory, in which a curve is a one-dimensional scheme and the theorem is a statement about the Euler characteristic of a coherent sheaf, is in Sheaves in Algebraic Geometry. The theory of the present article is the geometry over an algebraically closed field in the classical sense, which is the case the intuition is built on and the case to which the general theory must return. Nothing here uses a distance, a measure or a limit: the Zariski topology is an order-theoretic — that is, a purely algebraic — structure, and the topological vocabulary of the opening sections of Part II is used only in the sense of an arbitrary topological space of Topological Spaces, with no separation, compactness in the analytic sense, or metric available.

The article is organised as follows. The first section sets up affine space, the Zariski topology and the Nullstellensatz, with its dictionary between ideals and closed subsets. The second section studies the Zariski topology itself — irreducibility, the Noetherian property, the decomposition into irreducible components — and draws the consequences that distinguish it from the topologies of analysis. The third section treats morphisms and the equivalence of categories, the fourth projective varieties and the elimination-theoretic facts that make them complete, the fifth rational maps, function fields and birational geometry, and the sixth dimension, tangent spaces and smoothness. Throughout, $k$ is an algebraically closed field and the general base ring convention of the corpus is relaxed to this case as the classical theory requires, with the passage to a general ring, and to a field that is not algebraically closed, made.

Affine Varieties and the Nullstellensatz

Definition. Let $k$ be an algebraically closed field. Affine $n$-space over $k$ is the set

$$ \mathbb{A}^n_k = \{(a_1,\ldots,a_n) : a_i\in k\} = k^n $$

of $n$-tuples, and for a subset $S\subseteq k[x_1,\ldots,x_n]$ its zero set is

$$ V(S) = \{P\in\mathbb{A}^n_k : f(P) = 0 \text{ for all } f\in S\}. $$

For a subset $X\subseteq\mathbb{A}^n_k$ its vanishing ideal is

$$ I(X) = \{f\in k[x_1,\ldots,x_n] : f(P) = 0 \text{ for all } P\in X\}. $$

A subset $X$ is closed if $X = V(S)$ for some $S$, and the Zariski topology on $\mathbb{A}^n_k$ is the topology whose closed sets are these. A closed subset of $\mathbb{A}^n_k$ with the induced topology is an affine variety, and its coordinate ring is the quotient

$$ k[X] = k[x_1,\ldots,x_n]/I(X). $$

Theorem (the Hilbert basis theorem and the descending chain condition). The ring $k[x_1,\ldots,x_n]$ is Noetherian, and consequently the closed subsets of $\mathbb{A}^n_k$ satisfy the descending chain condition: every descending chain $X_1\supseteq X_2\supseteq\cdots$ of closed subsets is eventually constant. The Hilbert basis theorem is Part I's Noetherian and Artinian Rings; the consequence is that every closed set has a finite decomposition into irreducible ones.

Theorem (the Nullstellensatz). Let $k$ be algebraically closed and let $\mathrm{A}\subseteq k[x_1,\ldots,x_n]$ be an ideal.

Weak form: if $\mathrm{A}\neq(1)$ then $V(\mathrm{A})\neq\emptyset$; equivalently every maximal ideal of $k[x_1,\ldots,x_n]$ is of the form $(x_1-a_1,\ldots,x_n-a_n)$ and is therefore the vanishing ideal of a point.
Strong form: $I(V(\mathrm{A})) = \sqrt{\mathrm{A}}$, the radical of $\mathrm{A}$, the set of $f$ with $f^m\in\mathrm{A}$ for some $m\geq1$.

Proof. The weak form is Zariski's lemma: if $\mathrm{M}$ is maximal then $k[x_1,\ldots,x_n]/\mathrm{M}$ is a field finitely generated as a $k$-algebra, hence algebraic over $k$ by the lemma, hence equal to $k$, and the images of the $x_i$ give the point. The strong form follows by the Rabinowitsch trick: adjoining an indeterminate $t$ and applying the weak form to the ideal generated by $\mathrm{A}$ and $tf - 1$ produces the identity $f^m\in\mathrm{A}$. The proof of Zariski's lemma is the standard one, quoted here from Part I's Fields and Noetherian and Artinian Rings.

Corollary (the ideal–geometry dictionary). With $k$ algebraically closed, the maps $V$ and $I$ give inclusion-reversing bijections

$$ \begin{aligned} \{\text{radical ideals of } k[x_1,\ldots,x_n]\} &\longleftrightarrow \{\text{closed subsets of } \mathbb{A}^n_k\},\\ \{\text{prime ideals}\} &\longleftrightarrow \{\text{irreducible closed subsets}\},\\ \{\text{maximal ideals}\} &\longleftrightarrow \{\text{points}\}, \end{aligned} $$

and the same statements hold with $\mathbb{A}^n_k$ replaced by an affine variety $X$ and $k[x_1,\ldots,x_n]$ replaced by the coordinate ring $k[X]$: the radical ideals of $k[X]$ correspond to the closed subsets of $X$, the primes to the irreducible closed subsets and the maximal ideals to the points.

Proof. The first bijection is the Nullstellensatz together with the elementary facts $V(I(X)) = \overline{X}$ and $I(V(\mathrm{A})) = \sqrt{\mathrm{A}}$, and the containment-reversal of both maps. A closed set $V(\mathrm{A})$ is irreducible if and only if $\sqrt{\mathrm{A}}$ is prime: if $fg\in\mathrm{A}$ with neither $f$ nor $g$ in $\sqrt{\mathrm{A}}$, the nonvanishing of $f$ and $g$ on nonempty open sets of $V(\mathrm{A})$ makes both $V(f)\cap V(\mathrm{A})$ and $V(g)\cap V(\mathrm{A})$ proper closed subsets whose union is $V(\mathrm{A})$, and conversely a decomposition $V(\mathrm{A}) = V(\mathrm{B})\cup V(\mathrm{C})$ with neither part contained in the other produces a product in $\mathrm{A}$ with neither factor in $\sqrt{\mathrm{A}}$. The last item is the weak Nullstellensatz, and the statements for $X$ follow by applying the polynomial-ring version to a radical ideal defining $X$.

Corollary (the coordinate ring determines the variety). For affine varieties $X\subseteq\mathbb{A}^m_k$ and $Y\subseteq\mathbb{A}^n_k$ the following data determine one another: the variety $X$, its coordinate ring $k[X]$, and the set of $k$-algebra homomorphisms $k[X]\to k$ with the Zariski topology. In particular $k[X]$ is a finitely generated reduced $k$-algebra, and conversely every finitely generated reduced $k$-algebra is the coordinate ring of an affine variety, unique up to isomorphism.

Example (the dictionary in low dimension). The closed subsets of $\mathbb{A}^1_k$ are $\mathbb{A}^1_k$ itself and the finite sets, since $k[x]$ is a principal ideal domain; the Zariski topology on $\mathbb{A}^1_k$ is the cofinite topology, and on $\mathbb{A}^2_k$ the closed sets are $\mathbb{A}^2_k$, the finite unions of curves $V(f)$ for $f$ irreducible, and finite sets of points. The varieties $V(y^2 - x^3)$ and $V(y^2 - x^3 - x^2)$ in $\mathbb{A}^2_k$ are both irreducible plane curves of dimension one: the first has a cusp at the origin, and the second a node there when $k$ has characteristic not two, with two branches through the singular point instead of one. Over such a field the two curves are not isomorphic, the number of branches through a singular point being an invariant of the local ring, and that is a statement about the algebra rather than about the ambient plane. They are nevertheless birational: the parametrisations $(x,y) = (t^2,t^3)$ and $(x,y) = (t^2-1, t(t^2-1))$ exhibit the two coordinate rings as subrings of $k[t]$, so both function fields are the rational function field $k(t)$ and the two curves are birational while not being isomorphic. The characteristic does enter: when $k$ has characteristic two the substitution $y\mapsto y+x$ carries $y^2-x^3$ to $y^2+x^2-x^3 = y^2-x^3-x^2$, so the two curves, the node having coalesced with the cusp, are isomorphic there.

Example (the parabola and the coordinate ring). For $X = V(y - x^2)\subseteq\mathbb{A}^2_k$ the coordinate ring is $k[x,y]/(y-x^2)\cong k[x]$, and the projection $(x,y)\mapsto x$ is a bijection of varieties; $k[X]$ being a polynomial ring in one variable is the algebraic statement that $X$ is isomorphic to the affine line.

The Zariski Topology

Definition. A topological space $X$ is irreducible if it is nonempty and cannot be written as the union of two proper closed subsets; equivalently every nonempty open subset is dense. A Noetherian topological space is one whose closed subsets satisfy the descending chain condition. The dimension of a nonempty Noetherian space is the supremum of the lengths $n$ of chains $X_0\subsetneq X_1\subsetneq\cdots\subsetneq X_n$ of irreducible closed subsets.

Theorem (irreducibility and components). Let $X$ be an affine variety.

$X$ is irreducible if and only if $I(X)$ is prime, if and only if $k[X]$ is an integral domain.
$X$ is a Noetherian topological space, and it has a unique irredundant decomposition $$X = X_1\cup\cdots\cup X_r$$ into maximal irreducible closed subsets, its irreducible components, which are the closed sets $V(\mathrm{P})$ for the minimal primes $\mathrm{P}\supseteq I(X)$.
Every nonempty open subset of an irreducible $X$ is irreducible and dense; the open sets of the Zariski topology are, for irreducible $X$, exactly the complements of the proper closed subsets, each of which is dense.

Proof. (1) is the ideal-theoretic characterisation of the previous section. (2) the descending chain condition on closed sets gives a maximal irreducible closed subset inside any closed set, hence finitely many of them cover $X$, and an irredundant cover is unique; the correspondence with the minimal primes is the dictionary. (3) is the definition of irreducibility together with the fact that an open subset of an irreducible space is irreducible.

Theorem (properties of the Zariski topology). Let $X$ be an affine variety of positive dimension over the algebraically closed field $k$, with the Zariski topology.

$X$ is not Hausdorff: any two nonempty open subsets of an irreducible $X$ intersect, so the topology has no separation beyond $T_1$, and a nonempty open subset is dense.
$X$ is quasicompact, since the descending chain condition on closed sets makes every open cover contain a finite subcover.
Every open subset of $X$ is quasicompact, but $X$ is not in general Hausdorff and the compact subsets in the metric sense have no role: the topology is not metrisable when $\dim X>0$ and $k$ is uncountable.
The closure of a constructible set is closed, and a morphism of varieties carries constructible sets to constructible sets.

Proof. (1) If $U,V$ are nonempty open and disjoint, the complements are proper closed sets covering $X$, against irreducibility; in a $T_1$ space points are closed, and the only further separation available is that of distinct points by open sets, not by disjoint ones. (2) and (3) are the Noetherian property of the topology, which is inherited by the open subsets over a Noetherian ring. (4) is Chevalley's theorem on constructible sets, quoted as standard; the constructible subsets, finite unions of locally closed sets, are exactly the images of morphisms of varieties, the statement being the algebraic form of the elimination of quantifiers.

Remark (why the topology is so weak). The Zariski topology is designed so that the points of a variety can be recovered from the algebra of functions with no analytic input: the closed sets are the zero loci of finitely many polynomials, and there are so few of them that a nonempty open set is always dense in the irreducible case. Consequences recorded throughout the article follow: the constant sheaf has the wrong cohomology in this topology, a morphism need not have closed image, and the "compactness" of $X$ is the algebraic descending chain condition rather than a metric compactness. The compensating structure is the structure sheaf of Sheaves in Algebraic Geometry, whose sections over an open set are the regular functions, and which carries all the local information the topology omits.

Morphisms of Varieties

Definition. Let $X\subseteq\mathbb{A}^m_k$ and $Y\subseteq\mathbb{A}^n_k$ be affine varieties. A map $\varphi : X\to Y$ is a morphism of varieties if it is the restriction of a polynomial map, that is, $\varphi(P) = (f_1(P),\ldots,f_n(P))$ for polynomials $f_1,\ldots,f_n\in k[x_1,\ldots,x_m]$. A map is regular if it is a morphism in this sense, and an isomorphism if it is a bijective morphism whose inverse is a morphism.

Theorem (the contravariant equivalence). The assignment $X\mapsto k[X]$ extends to an equivalence of categories

$$ \{\text{affine varieties over } k\}^{\mathrm{op}}\ \xrightarrow{\ \sim\ }\ \{\text{finitely generated reduced } k\text{-algebras}\}, $$

under which a morphism $\varphi : X\to Y$ corresponds to the $k$-algebra homomorphism $\varphi^* : k[Y]\to k[X]$, $\varphi^*(g) = g\circ\varphi$. In particular $\varphi$ is an isomorphism if and only if $\varphi^*$ is, and the assignment is fully faithful.

Proof. A polynomial map $\mathbb{A}^m_k\to\mathbb{A}^n_k$ pulls polynomial functions back to polynomial functions, and the pullback descends to the coordinate rings because $\varphi(X)\subseteq Y$ forces $\varphi^*(I(Y))\subseteq I(X)$. Conversely a $k$-algebra homomorphism $k[Y]\to k[X]$ corresponds by the universal property of the quotient to a polynomial map $X\to Y$: the images of the coordinate functions of $Y$ define the components of a polynomial map, and the condition that $I(Y)$ map into $I(X)$ is exactly the condition that the image lie in $Y$. The two constructions are inverse, and the remaining statements are formal from the bijection on hom-sets and the identification of the coordinate ring of $\mathbb{A}^n_k$.

Corollary (products and graphs). The product of affine varieties $X$ and $Y$ exists, and

$$ k[X\times Y] = k[X]\otimes_k k[Y], $$

the tensor product over $k$; the diagonal $\Delta : X\to X\times X$ and the graph $\Gamma_\varphi\subseteq X\times Y$ of a morphism are closed subvarieties, the graph being the zero set of the equations $y_i = \varphi_i(x)$. Consequently the categorical product of affine varieties, and the fibre product over a point, are affine varieties.

Proof. The product $\mathbb{A}^m_k\times\mathbb{A}^n_k = \mathbb{A}^{m+n}_k$ has coordinate ring $k[x_1,\ldots,x_m]\otimes_kk[y_1,\ldots,y_n]$ by the universal property of the tensor product of polynomial rings; the subvariety $X\times Y$ is defined by the ideal generated by $I(X)$ and $I(Y)$, and for a radical ideal $I(X)$ and a perfect field $k$ the tensor product $k[X]\otimes_kk[Y]$ is reduced, so it is the coordinate ring of $X\times Y$. The diagonal and the graph are closed since $x_i-y_i$ and $y_i-\varphi_i(x)$ are polynomials.

Example (morphisms to the affine line and to the punctured line). A morphism $X\to\mathbb{A}^1_k$ is exactly an element of $k[X]$, by the equivalence; a morphism $X\to\mathbb{A}^1_k\setminus\{0\}$ is a unit of $k[X]$, so for $X = \mathbb{A}^1_k$ there are no nonconstant morphisms from the affine line to the punctured affine line, in sharp contrast with the analytic or the metric situation. The statement is the algebraic beginning of the divisor theory of Part I's Algebraic Curves.

Projective Varieties

Definition. Projective $n$-space over $k$ is the set

$$ \mathbb{P}^n_k = \bigl(\mathbb{A}^{n+1}_k\setminus\{0\}\bigr)\big/k^\times $$

of lines through the origin, written in homogeneous coordinates $[a_0:\cdots:a_n]$; for a homogeneous polynomial $F\in k[x_0,\ldots,x_n]$ the zero set

$$ V(F) = \{[a]\in\mathbb{P}^n_k : F(a_0,\ldots,a_n) = 0\} $$

is well defined because $F(\lambda a) = \lambda^{\deg F}F(a)$, and a projective variety is the zero set in $\mathbb{P}^n_k$ of a set of homogeneous polynomials with the induced Zariski topology. Writing $U_i = \{[a] : a_i\neq0\}\cong\mathbb{A}^n_k$ with coordinates $x_j/x_i$, the $n+1$ open sets $U_i$ cover $\mathbb{P}^n_k$, and the homogeneous coordinate ring of a projective variety $X\subseteq\mathbb{P}^n_k$ is

$$ S(X) = k[x_0,\ldots,x_n]/I(X), \qquad I(X) = \{F \text{ homogeneous} : F|_X = 0\}, $$

a graded ring over $k$ whose homogeneous ideals define the closed subsets of $X$.

Theorem (the projective Nullstellensatz and the affine charts). Let $k$ be algebraically closed.

The maps $V$ and $I$ give inclusion-reversing bijections between the nonempty closed subsets of $\mathbb{P}^n_k$ and the saturated homogeneous radical ideals $\mathrm{A}$ of $k[x_0,\ldots,x_n]$ with $\mathrm{A}\neq(x_0,\ldots,x_n)$, where the saturation of a homogeneous ideal is $\bigcup_{m\geq0}(\mathrm{A}:(x_0,\ldots,x_n)^m)$; the zero set in $\mathbb{P}^n_k$ depends only on the saturation, and $V(\mathrm{A}) = V(\sqrt{\mathrm{A}})$ for homogeneous $\mathrm{A}$.
$X\cap U_i$ is an affine variety with coordinate ring the degree-zero part $S(X)_{(x_i)}$ of the localisation of the homogeneous coordinate ring at $x_i$; the passages $X\cap U_i = X\cap U_j$ glue the charts, and a closed subset of $\mathbb{P}^n_k$ is irreducible if and only if its homogeneous vanishing ideal is prime.
The image of a morphism from a projective variety to any variety is closed: projective varieties are complete.

Proof. (1) and (2) are the projective forms of the Nullstellensatz and of the localisation of the homogeneous coordinate ring in the cover by the $U_i$, both standard; the graded version of the radical correspondence is proved as in the affine case, the irrelevant ideal being the only one with empty zero set. (3) is the fundamental theorem of elimination theory: a morphism $\varphi : X\to Y$ with $X$ projective is given by homogeneous polynomials on the charts, and the image is the zero set of the resultants obtained by eliminating the homogeneous coordinates; the proof is the elimination theory of Part I's Gröbner Bases and Elimination Theory.

Example (the Segre and Veronese embeddings). The Segre embedding

$$ \sigma : \mathbb{P}^m_k\times\mathbb{P}^n_k\to\mathbb{P}^{mn+m+n}_k, \qquad ([a],[b])\mapsto[a_ib_j], $$

is a closed embedding whose image is the variety defined by the vanishing of all the $2\times2$ minors of the matrix $(z_{ij})$, and it exhibits the product of projective varieties as a projective variety; the Veronese embedding

$$ \nu_d : \mathbb{P}^n_k\to\mathbb{P}^{N}_k, \qquad [a]\mapsto[\text{all monomials of degree } d], \qquad N = \binom{n+d}{n}-1, $$

is a closed embedding whose image is a linearly normal variety, and it linearises the hypersurfaces of degree $d$: the $d$-fold hypersurfaces in $\mathbb{P}^n$ correspond to the hyperplane sections of the Veronese variety. Both are standard examples of the projective constructions that replace the products and the quotients unavailable in the affine category.

Remark (projective versus affine). A projective variety is complete, and an affine variety of positive dimension is never complete; the standard proof that $\mathbb{A}^1_k$ is not complete is the existence of the nonconstant nonclosed-image morphism $\mathbb{A}^1_k\to\mathbb{A}^1_k\setminus\{0\}$ of the previous section. Completeness is the algebraic counterpart of compactness, and it is an algebraic condition — the closedness of the images of morphisms — not a topological one: in the Zariski topology every variety already satisfies the finite-subcover property, so the two notions of compactness do not coincide and the algebraic one is the useful one. The general treatment of properness is in Schemes.

Rational Maps, Function Fields and Birational Geometry

Definition. Let $X$ be an irreducible affine or projective variety. The function field $k(X)$ is the fraction field of the coordinate ring $k[X]$ in the affine case, and in the projective case the field of fractions of the degree-zero part of the localisation of $S(X)$ at a coordinate, the field being independent of the chart; its elements are the rational functions on $X$. A rational map $\varphi : X\dashrightarrow Y$ is an equivalence class of morphisms $U\to Y$ from nonempty open subsets $U\subseteq X$, two such being equivalent if they agree on a nonempty open subset. A rational map is dominant if its image is dense in $Y$; it is birational if it has a rational inverse, and $X,Y$ are then birational or birationally equivalent.

Theorem (the equivalence of categories of function fields). Let $X,Y$ be irreducible varieties over $k$. Pulling back a rational function along a dominant rational map induces an embedding $k(Y)\hookrightarrow k(X)$ of fields, and the assignment is a contravariant equivalence between the category of irreducible varieties over $k$ with dominant rational maps and the category of finitely generated field extensions of $k$, of transcendence degree $\dim X$. In particular:

$X$ and $Y$ are birational if and only if $k(X)\cong k(Y)$ over $k$;
the dimension of $X$ is the transcendence degree of $k(X)$ over $k$, and also the Krull dimension of $k[X]$, which is the maximal length of a chain of irreducible closed subvarieties of $X$;
the inclusion $k[X]\subseteq k(X)$ realises $X$ up to birational equivalence; the local rings of $X$ are the localisations of $k[X]$ at primes, and the points of codimension one are exactly those whose local ring is a discrete valuation ring of $k(X)$, so that the Weil divisors on $X$ are the free abelian groups on the discrete valuations of $k(X)$ trivial on $k$, as in Part I's Valuation Theory and Henselian Rings and Algebraic Curves.

Proof. A dominant rational map is represented on a nonempty open set, and the pullback of a rational function on $Y$ is a rational function on $X$; the assignment is functorial and injective by dominance. The inverse assignment sends a finitely generated field extension $K/k$ to the variety of a maximal set of algebraically independent generators, the coordinate ring being the $k$-subalgebra generated by them, and the two assignments are mutually inverse as the equivalence of the previous section is on the level of the rings. The claim on dimension is the dimension theorem for finitely generated algebras over a field, Part I's Integral Extensions and Krull Dimension; the claim on the points and the valuations is Part I's Valuation Theory and Henselian Rings combined with the local description of the structure sheaf.

Theorem (blow-ups and the resolution of singularities). Let $X$ be an irreducible variety and $Z\subseteq X$ a closed subvariety.

The blow-up of $X$ along a closed subvariety $Z$ of codimension at least one exists: a variety $\tilde X$ with a morphism $\pi : \tilde X\to X$ which is an isomorphism over $X\setminus Z$ and such that the inverse image of $Z$ is a divisor on $\tilde X$; when $X = \mathbb{A}^n_k$ and $Z = \{0\}$, the blow-up is $$\tilde X = \{(a,[b])\in\mathbb{A}^n_k\times\mathbb{P}^{n-1}_k : a_ib_j = a_jb_i \text{ for all } i,j\},$$ which is the closure of the graph of the rational map $a\mapsto[a]$ and is an isomorphism over $\mathbb{A}^n_k\setminus\{0\}$.
Resolution of singularities: for a variety over a field of characteristic zero there is a proper birational morphism $\pi : \tilde X\to X$ with $\tilde X$ smooth, obtained by a finite sequence of blow-ups along smooth centres. This is Hironaka's theorem, quoted as the standard result; it is an existence theorem in the birational category and it is not constructive in the present article.

Proof. (1) The equations $a_ib_j = a_jb_i$ define a closed subvariety of $\mathbb{A}^n\times\mathbb{P}^{n-1}$, and the projection to $\mathbb{A}^n$ is an isomorphism away from the origin because $a\neq0$ determines $[b] = [a]$; over the origin the fibre is all of $\mathbb{P}^{n-1}$, the exceptional divisor. (2) is quoted.

Example (birationally equivalent but non-isomorphic varieties). The nodal cubic $C = V(y^2 - x^3 - x^2)\subseteq\mathbb{A}^2_k$ and the affine line are birational and not isomorphic: the rational map $\mathbb{A}^1_k\dashrightarrow C$, $t\mapsto(t^2-1,t(t^2-1))$, is birational with inverse $x/y$ on the complement of the node, and the coordinate rings $k[x,y]/(y^2-x^3-x^2)$ and $k[t]$ are not isomorphic, since the former has a non-free class group and the latter a unit group $k^\times$ and the former also; the invariants that distinguish them are the local rings at the singular point, which are not regular. The example is the first case of the principle that birational geometry is the geometry of function fields, and that the passage from a birational class to its members is the classification of models.

Dimension, Tangency and Smoothness

Definition. Let $X\subseteq\mathbb{A}^n_k$ be an affine variety and $P\in X$. The tangent space is

$$ T_PX = \{v\in k^n : \sum_i \frac{\partial f}{\partial x_i}(P)\,v_i = 0 \text{ for all } f\in I(X)\}, $$

the common kernel of the differentials of the defining equations, and $P$ is a smooth or nonsingular point of $X$ if $\dim_kT_PX = \dim X$; otherwise $P$ is singular, and $X$ is smooth if every point is smooth. The singular locus $\operatorname{Sing}(X)$ is the set of singular points.

Theorem (dimension and the tangent space). Let $X$ be an irreducible affine variety of dimension $d$.

$\dim_kT_PX\geq d$ for every $P$, and $P$ is smooth if and only if the local ring $\mathcal{O}_{X,P}$ is a regular local ring, if and only if the maximal ideal $\mathrm{M}_P$ satisfies $\dim_k\mathrm{M}_P/\mathrm{M}_P^2 = d$; the tangent space is dual to $\mathrm{M}_P/\mathrm{M}_P^2$.
The singular locus is a proper closed subset of $X$: it is the zero set of the $d\times d$ minors of the Jacobian matrix of a set of generators of $I(X)$.
The set of smooth points is a nonempty dense open subset of $X$: $X$ is smooth if and only if every local ring $\mathcal{O}_{X,P}$ is regular.

Proof. (1) The inequalities and the identification of the tangent space with the dual of $\mathrm{M}_P/\mathrm{M}_P^2$ are the standard local computation: the differentials of the defining equations generate the relations and the rank of the Jacobian is at most the codimension, so the tangent dimension is at least $d$ and equality is exactly regularity of the local ring. (2) the rank condition is expressed by the vanishing of the maximal minors, which are polynomials, so the singular locus is closed; it is proper because the field being perfect makes the extension $k(X)/k$ separably generated, so $X$ is birational to a hypersurface in $\mathbb{A}^{d+1}$ whose defining equation, being squarefree, has a nonvanishing gradient on a nonempty open set; the smooth locus is therefore nonempty for one model of the function field, and the nonemptiness of the smooth locus is a birational invariant. (3) is (2) together with the irreducibility of $X$, which makes every nonempty open subset dense.

Example (singularities of plane curves). For an irreducible plane curve $C = V(f)\subseteq\mathbb{A}^2_k$ with $f$ squarefree, a point $P$ is singular if and only if $f(P) = f_x(P) = f_y(P) = 0$; the nodal cubic of the previous examples has exactly one singular point, the origin, where the tangent space is two-dimensional while the curve has dimension one, and the cuspidal cubic $V(y^2-x^3)$ likewise; the smooth conic $V(xy-1)$ has no singular point. The multiplicity of $P$ on the curve is the lowest degree of a nonzero term of the Taylor expansion of $f$ at $P$, and it is $1$ exactly at the smooth points.

Corollary (the Jacobian criterion). Let $X = V(f_1,\ldots,f_r)\subseteq\mathbb{A}^n_k$ be an irreducible variety of dimension $d$. Then $X$ is smooth at $P$ if and only if the Jacobian matrix $\bigl(\partial f_i/\partial x_j(P)\bigr)$ has rank $n-d$; if the rank is $n-d$ at every point of $X$, then $I(X) = (f_1,\ldots,f_r)$ and $X$ is a complete intersection. The criterion is the practical test for smoothness and it is the algebraic form of the implicit function theorem of Smooth Manifolds and Differential Geometry; the analytic theorem itself, which requires a distance and a limit, is Part III's.

Summary

An affine variety over an algebraically closed field $k$ is a closed subset of $\mathbb{A}^n_k$ in the Zariski topology, and its geometry is the algebra of its coordinate ring $k[X] = k[x_1,\ldots,x_n]/I(X)$: the Nullstellensatz makes the correspondence between closed subsets and radical ideals a bijection, and its weak form identifies the points with the maximal ideals and its strong form identifies the closed subsets with the radical ideals. The dictionary extends to the irreducible closed subsets, which correspond to the prime ideals and to the irreducible components of a Noetherian topological space, and to the dimension, which is at once the Krull dimension of the coordinate ring, the transcendence degree of the function field, and the maximal length of a chain of irreducible closed subvarieties. The Zariski topology is quasicompact but not Hausdorff, and on an irreducible variety every nonempty open subset is dense; the compensating structure is the structure sheaf, whose sections are the regular functions, and the sheaves of modules over it, treated in Sheaves in Algebraic Geometry.

Morphisms of affine varieties are the polynomial maps, and they are contravariantly equivalent to the homomorphisms of coordinate rings, so that the category of affine varieties is the opposite of the category of finitely generated reduced $k$-algebras; products exist and correspond to tensor products, and graphs and diagonals are closed. Projective varieties are the closed subsets of projective space defined by homogeneous polynomials, their homogeneous coordinate rings are graded, and they are complete: the image of any morphism from a projective variety is closed, by the elimination theory behind the fundamental theorem of elimination theory. Blow-ups and the resolution of singularities belong to birational geometry, whose invariant is the function field: a dominant rational map pulls back rational functions, irreducible varieties with dominant rational maps are contravariantly equivalent to finitely generated field extensions of $k$, and birational equivalence is isomorphism of function fields. Finally, the tangent space at a point is dual to $\mathrm{M}_P/\mathrm{M}_P^2$, smoothness is the equality of its dimension with $\dim X$ and is equivalent to the regularity of the local ring, the singular locus is a proper closed subset, and the Jacobian criterion is the effective form of the implicit function theorem in the algebraic setting. The scheme-theoretic generalisation, dropping the algebraic closedness, the reducedness and the embedding, is; the curve theory is Part I's.

Summary of Notation

Symbol	Meaning
$k$	algebraically closed field, the base of the classical theory
$\mathbb{A}^n_k$, $\mathbb{P}^n_k$	affine and projective $n$-space; $U_i$ the standard affine charts
$V(S)$, $I(X)$	zero set of a set of polynomials; vanishing ideal of a subset
$k[X] = k[x_1,\ldots,x_n]/I(X)$	coordinate ring of an affine variety
$S(X)$, $I(X)$	homogeneous coordinate ring and its homogeneous vanishing ideal
Zariski topology	closed sets are the $V(\mathrm{A})$; not Hausdorff, quasicompact, Noetherian
irreducible, component	not a union of two proper closed subsets; maximal irreducible closed subsets
morphism, $\varphi^*$	polynomial map; pullback $k[Y]\to k[X]$
$k(X)$	function field; fraction field of $k[X]$; transcendence degree $\dim X$
rational map, birational	morphism on a dense open set up to agreement; inverse rational map
blow-up	$\tilde X\to X$ isomorphism over $X\setminus Z$ with inverse image of $Z$ a divisor
$T_PX$, smooth, singular	tangent space; $\dim_kT_PX = \dim X$; $\operatorname{Sing}(X)$ closed and proper
dimension	Krull dimension of $k[X]$ = transcendence degree of $k(X)$ = chain length