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A classical variety is affine if finitely many global regular functions generate the unit ideal and their nonvanishing opens are affine. Localization of global sections on a principal open identifies their coordinate rings with $A_{f_i}$, where $A=\Gamma (X,{\mathcal{O}}_X)$. Choose a finite-type subalgebra of $A$ containing the functions, their unit-ideal coefficients and numerators of generators of every $A_{f_i}$. Its affine variety has the same principal affine charts, whose isomorphisms glue globally. This avoids assuming finite generation of $A$ before affineness is known.

An algebraic variety over a field $k$ is, in a common modern convention, a reduced scheme that is a separated scheme of finite type over $k$. Some authors also require it to be irreducible; this wiki states that extra condition when reducible spaces matter. A classical convention over an algebraically closed field describes the same geometry using polynomial zero sets and their regular functions.

An affine variety has a coordinate ring $A$, a finitely generated reduced $k$-algebra, and its scheme model is $\mathrm{Spec}A$. General algebraic varieties glue finitely many such affine charts along open subsets. The regular functions are part of the structure: their local rings record which functions can be evaluated near each point.

For example, $x^2+y^2=1$ defines a real algebraic curve, while $y^2=x^3$ defines a cuspidal cubic. Both are one-dimensional, but the cusp has a singular local ring at the origin. The scheme defined by $x^2=0$ additionally remembers a nonzero nilpotent element; it is not a reduced scheme and so is outside this variety convention. This illustrates why an underlying zero set alone cannot describe every scheme.

For an integral normal variety that is Noetherian, removing a closed subset of codimension at least two does not create new regular functions. Affine-locally, a normal Noetherian domain is the intersection of its localizations at height-one primes inside its fraction field. The complement still contains every height-one point, so a regular function on it lies in each such localization and therefore in the original ring. On projective space this also follows directly by excluding irreducible polynomial factor of the denominators on each polynomial affine chart.

A complex algebraic variety is an algebraic variety over the complex numbers. In the scheme convention used here, it is a reduced separated scheme of finite type over $\mathbb{C}$; irreducibility is an additional condition when needed. Affine examples are zero sets of complex polynomials equipped with their regular functions.

For example, $xy=0$ is the union of two complex coordinate lines, while $y^2=x^3$ is an irreducible singular curve. The smooth locus of a complex variety has the structure of a complex manifold, but a singular point need not be smooth as a complex analytic space.

One must also distinguish topologies. The complex affine line is $\mathbb{C}$ as a point set; its proper Zariski-closed subsets are finite, whereas its usual analytic topology has many more closed and open subsets. Passing from an algebraic variety to its analytic space retains geometric information while changing which local functions and open sets are allowed.

Two nonzero elements with no common nonunit irreducible factor. In a unique factorization domain this is equivalent to their greatest common divisor being a unit. Coprimality of numerator and denominator makes a fraction reduced, so simultaneous restrictions on its denominator can force it to be a regular function.

A regular function $f$ on an entire affine variety is a polynomial function, even if the variety is reducible. Put $J=\{g\in A(X):gf\in A(X)\}$, an ideal of its coordinate ring. Near a point $p$, write $f=a/b$ with $b$ nonzero, and choose a principal open subset $D(h)$ containing $p$ inside that neighbourhood. Then $(hb)f=ha$ on all of $X$: on $D(h)$ this is the fraction identity, and off $D(h)$ both sides vanish. Hence $hb\in J$ and $(hb)(p)\ne 0$. No point is a common zero of $J$, so the weak Hilbert Nullstellensatz implies $J=A(X)$. Thus $1\in J$ and $f\in A(X)$. This proof does not require a fraction field for a reducible coordinate ring.

On an integral scheme $X$, every nonempty open $U$ contains its generic point $\eta$. Taking a germ gives an injective ring homomorphism from its regular functions to the function field ${\mathcal{O}}_{X,\eta }$. Injectivity follows on each nonempty affine open subscheme from the injection of an integral domain into its field of fractions, and then from the sheaf gluing axiom. This lets regular functions on different open subsets be compared inside one field.

For an algebraic group $G$ acting on a algebraic variety $X$ over an algebraically closed field, a geometric quotient is a surjective invariant morphism $q:X\to Y$ whose fibers are exactly the group orbits, whose topology is the quotient topology, and whose regular functions are the invariant regular functions locally:The orbit condition gives actual orbit parameters; the function-sheaf condition specifies the algebraic structure on the orbit space. Both matter beyond a set-theoretic quotient.

For scalar multiplication on ${\mathbb{A}}^{n+1}\setminus \{0\}$, the map $x\mapsto [x]$ is the geometric quotient by ${\mathbb{G}}_m$, giving projective space. On a standard chart, coordinate ratios are the invariant functions. In the Cox construction, the quotient is geometric for a simplicial toric fan; in general the categorical quotient can identify more than a single orbit.^[ref]

Over an algebraically closed field, every regular function on a nonempty irreducible projective variety is constant. Regard the function as a morphism to ${\mathbb{P}}^1$ avoiding infinity. Its graph is closed; the closedness of projection from projective space makes its image closed in ${\mathbb{P}}^1$. A proper closed subset there is finite, and an irreducible finite image is a singleton. Irreducibility is the usual variety convention; on a disconnected projective algebraic set different components may have different constants.

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.

For distinct closed points over an algebraically closed field, evaluation gives the short exact sequence $0\to {\mathcal{I}}_{\{P,Q\}}\to {\mathcal{O}}_X\to k_P\oplus k_Q\to 0$. If all global regular functions are constant, the long exact sequence in sheaf cohomology embeds $k^2/k(1,1)$ into $H^1(X,{\mathcal{I}}_{\{P,Q\}})$. Thus this ideal detects a concrete obstruction to affineness.

For a quasi-compact separated scheme and a global regular function $f$, global sections on $X_f$ are the localization of global sections by $f$. Choose a finite affine open cover. Its intersections are affine, so the sheaf gluing axiom describes global sections as the kernel of the difference map between two finite products of affine coordinate rings. Exactness of localization preserves this kernel, and localization commutes with those finite products. Localizing the same cover gives the claimed isomorphism. This applies before knowing whether $X$ itself is affine.

A matrix minor is the determinant of a square submatrix obtained by selecting rows and columns of a matrix. Over a field, the matrix rank is less than $r$ exactly when every $r$-by-$r$ matrix minor vanishes. In a matrix of regular functions, these equations define a closed rank-defect locus, explaining the determinantal variety construction.

A morphism $F:X\to Y$ is a continuous map in the Zariski topologies such that for every open $V\subseteq Y$ and every regular function $h$ on $V$, the pullback $h\circ F$ is regular on $F^{-1}(V)$.

Equivalently, near every point, choose affine charts of source and target; the target coordinates of $F$ must then be regular functions of the source coordinates. These local conditions agree on overlaps. In particular a morphism is defined at every point of its source, not merely on a dense open subset.

A regular function on $U$ is a function $f:U\to k$ locally representable, in an affine chart of the ambient projective space, as a quotient $a/b$ of polynomial functions with denominator nonzero on that neighbourhood. Equivalently it is locally a quotient of homogeneous forms of the same degree. The equal-degree condition makes the quotient independent of the chosen projective representative.

The ring ${\mathcal{O}}_{X,p}$ is the ring of germs of such functions near $p$. Two representatives $(U,f)$ and $(V,g)$ define the same germ when they agree on some smaller neighbourhood of $p$. Addition and multiplication use a common neighbourhood. It is a local ring: its unique maximal ideal consists of germs vanishing at $p$, and any germ nonzero at $p$ has a regular reciprocal on a smaller neighbourhood.

Put $\mathcal{D}={\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$, the quotient sheaf of abelian groups. At a closed point $P$, the discrete valuation gives an isomorphismIndeed, a local parameter on a smooth algebraic curve has valuation one, and the valuation kernel is precisely the unit group of the discrete valuation ring. Thus a local section represented by $f$ determines its local principal divisor, and changing $f$ by a regular function which is a unit does not change those coefficients.

These local valuations identify $\Gamma (X,\mathcal{D})$ with $\mathrm{Div}(X)$. To see that they give finite support, represent a section locally by finitely many rational functions on an algebraic variety, using Noetherian Zariski topology to take a finite cover. Each representative has only finitely many zeros and poles, so their combined support is finite. Conversely, for $D=\sum n_{P}P$, choose at each support point a local parameter $t_P$ and shrink its neighbourhood to exclude every other zero or pole of $t_P$ and every other support point. The local rational equation $t_P^{n_P}$ represents the desired valuations there; on the complement of the support use $1$. On overlaps these equations differ by units, so their classes glue in the quotient sheaf. Finally, a section with every valuation zero has zero germ everywhere and is the identity section. Hence the identification is both surjective and injective.

The map on global sections sends $f\in k(X{)}^{\ast }$ to $\mathrm{div}(f)$. Therefore its cokernel is

For a sheaf of abelian groups $\mathcal{F}$ on a topological space $X$, putThese arbitrary families of germs form a sheaf, and restrictions are projections. Extending a family by zero proves that $C(\mathcal{F})$ is flasque. Sending a section to all its germs gives an injective sheaf map $\mathcal{F}\to C(\mathcal{F})$, since a section with every germ zero vanishes locally and hence globally.

Set $Q^{-1}=\mathcal{F}$, $I^j=C(Q^{j-1})$ and $Q^j=\mathrm{coker}(Q^{j-1}\to I^j)$, using the sheaf cokernel. Compose $I^j\to Q^j\to I^{j+1}$ to obtain the Godement resolutionIt is exact by construction, with flasque terms. Define sheaf cohomology by the cohomology of its global-section complex:The resolution principle for sheaf cohomology identifies this with the computation using any flasque resolution.

The elementary input for acyclicity is the flasque-kernel section-lifting lemma: in a short exact sequence with flasque kernel, sections of the quotient lift over every open subset. Consequently a quotient of two flasque sheaves is flasque, because a quotient section lifts and that lift extends. If $\mathcal{F}$ itself is flasque, induction through $0\to Q^{j-1}\to I^j\to Q^j\to 0$ makes every $Q^j$ flasque. The same lifting lemma makes these sequences exact on global sections. Hence the global-section complex has no positive-degree cohomology, provingThis deduction uses the lifting lemma, not the desired cohomological vanishing as an assumption.

For the rational-section construction, add two representative sections after restricting to the intersection of their dense domains. Multiply by a representative rational function in the same way. Finite intersections of dense open sets are dense, and further restricting both representatives does not change their resulting class. The module axioms hold on their common domain. Thus rational sections form a module over $\mathrm{Rat}(X)$; irreducibility is not needed for this module statement. When $X$ is irreducible, $\mathrm{Rat}(X)$ is the function field $K=k(X)$.

Suppose now that $\mathcal{F}$ is locally free and $X$ irreducible. A germ at $P$ is represented on a nonempty open neighbourhood, which is dense, so it defines a rational section. Two representatives of the germ agree on a neighbourhood of $P$ and therefore give the same class. If the class is zero, trivialize $\mathcal{F}$ on a neighbourhood of $P$. Its coefficient functions vanish on a dense open subset; regular functions on a reduced irreducible variety with this property vanish identically. The germ is therefore zero. This proves the stalk inclusion into rational sections of a locally free sheaf:Local freeness matters: a nonzero torsion germ can vanish on a dense open set.

On an irreducible algebraic curve, every proper closed subset is finite, and every open subset is quasi-compact because the Zariski space is Noetherian. Let $R=\mathrm{Rat}(\mathcal{F})$. The constant sheaf $\mathcal{R}(\mathcal{F})$ has sections $R$ on every nonempty open set: such opens are irreducible and hence connected. DefineRestriction drops the components outside the smaller open set. For a compatible family over an open cover of $U$, take a finite subcover by quasi-compactness. The union of the finite supports of these finitely many sections is finite. Their agreeing components therefore glue to a unique finite-support tuple on $U$. This verifies the sheaf gluing axiom, including arbitrary covers, rather than merely asserting that a presheaf direct sum is always a sheaf.

A rational section is regular on some dense open set, whose complement on a curve is finite. Its classes $[s{]}_P\in R/{\mathcal{F}}_P$ therefore have finite support, defining a sheaf map $\mathcal{R}(\mathcal{F})\to \mathcal{P}(\mathcal{F})$. At $P$, the first sheaf has stalk $R$. The second has stalk $R/{\mathcal{F}}_P$: any finite tuple can be restricted to a neighbourhood omitting all its other support points, and its $P$-component is unaffected by such restriction. The resulting sequence on stalks isExactness of sheaves can be tested on stalks, so this proves the rational principal-parts resolution on an algebraic curveBoth right-hand sheaves are flasque: restrictions for the constant rational-section sheaf are identities between nonempty opens, and restrictions for the rational principal-parts sheaf are projections, with extension by zero. ConsequentlySurjectivity on stalks has not been confused with surjectivity on global sections.

For an explicit failure of global surjectivity take $X={\mathbb{P}}^1$ and $\mathcal{F}=\mathcal{O}(-2)$. Use coordinate $z=X_1/X_0$ and frame $e_0=X_0^{-2}$ on the finite chart. At infinity use $w=1/z$ and $e_{\infty }=X_1^{-2}=z^{-2}{e}_0$. Prescribe the rational principal part of $z^{-1}{e}_0$ at zero and zero at every other point. If a rational section $r(z)e_0$ realized this tuple, then $r-z^{-1}$ would be regular at zero and $r$ would be regular at every other finite point. Thus $r=z^{-1}+p(z)$ with $p\in k[z]$. Regularity at infinity would requireto be regular at $w=0$. Its $w^{-1}$ term cannot cancel with any term of the polynomial contribution, whose powers are at most $-2$. This is impossible. Hence the global rational-principal-part map need not be surjective.

For an affine curve it is always surjective, and here is a direct construction. Write $X=\mathrm{Spec}A$, $K=\mathrm{Frac}A$ and $\mathcal{F}=\tilde{M}$. By the affine module sheaf construction, $R=M{\otimes }_{A}K=:M_K$ and ${\mathcal{F}}_P=M_{{\mathfrak{m}}_P}$. Choose representatives $v_P\in M_K$ for a prescribed finite tuple, and choose one nonzero $a\in A$ with $a{v}_P\in M$ for all of them. The zero set $T=V(a)$ is finite. The ring $A/(a)$ is Artinian, since it is Noetherian of dimension zero, and its Artinian decomposition into local factors givesPrescribe the class of $a{v}_P$ in the indicated component for each requested point in $T$, and zero in the other components. A requested point outside $T$ already has zero principal part because $a$ is a unit there. Lift the tuple to $w\in M$. Then $s=w/a\in M_K$ satisfies $s-v_P\in M_{{\mathfrak{m}}_P}$ at every requested point in $T$, and is regular at every other point of $T$. Outside $T$ it is regular because $a$ is invertible. It therefore realizes exactly the prescribed tuple. This proves affine principal-parts interpolation for a locally free sheaf and givesNo nonsingularity assumption was used; the construction applies to singular irreducible affine curves as well.

On the standard affine charts $U_i=\{X_i\ne 0\}$, use the formal frame $e_i=X_i^m$ and glue rank-one free sheaves byThe transition functions are regular units on overlaps and satisfy the cocycle identity. This constructs the twisting sheaf on projective space $\mathcal{O}(m)$ as an invertible sheaf for every integer $m$, including negative $m$.

On a nonempty $U$, a section has local expressions $h_i{e}_i$, where $h_i\in \mathcal{O}(U\cap U_i)$. Pulling to the punctured affine cone gives $h_i(X/X_i)X_i^m$. Their agreement on overlaps makes them one homogeneous rational function $r$ of degree $m$, regular on ${\pi }^{-1}U$. To make its polynomial representation explicit, write one nonzero local coefficient as $p(y)/q(y)$ in the affine coordinates. Homogenize $p,q$ to degrees $a,b$; the resulting rational expression isIf the exponent of $X_i$ is negative, move its power to the denominator. Cancel common factors; the greatest common divisor of homogeneous polynomials can be chosen homogeneous. Thus $r=F/G$ with $F,G$ coprime homogeneous polynomials, $G\ne 0$, and $\mathrm{deg}F-\mathrm{deg}G=m$.

Conversely, for such a homogeneous rational function regular on ${\pi }^{-1}U$, its coefficient $r/X_i^m$ is regular on $U\cap U_i$: restrict to the slice $X_i=1$. These coefficients obey the displayed transition rule and hence define a section. This proves the homogeneous rational sections of a twisting sheaf description in both directions, for arbitrary open $U$ rather than only a standard chart.

Now lift a regular degree-zero rational function $f=P/Q$ to the punctured cone. The quotient rule givesThis is a rational function homogeneous of degree $-1$. It is independent of the chosen representation because differentiation is a derivation of the rational function field. It is regular wherever $f$ is regular: a derivation of a polynomial ring extends to each localization by the quotient rule, and regular functions are locally such fractions. The preceding homogeneous-section description therefore provesZeros of a particular displayed denominator do not invalidate this argument; regularity is a property of the rational function, which can have another local representation.

The map $f\mapsto ({\partial }_{0}f,\dots ,{\partial }_{n}f)$ is a $k$-linear sheaf derivation with values in $\mathcal{O}(-1{)}^{\oplus (n+1)}$. The universal property of Kähler differentials consequently definesMultiplication by the coordinate section $X_i\in \Gamma ({\mathbb{P}}^n,\mathcal{O}(1))$ defines the other mapFor a degree-$d$ homogeneous polynomial, differentiating each monomial gives ${\sum }_i{X}_i{\partial }_{i}P=dP$. Applying this to the equal-degree numerator and denominator gives ${\sum }_i{X}_i{\partial }_{i}f=0$ for degree-zero $f$, so $\sigma D=0$. This identity is valid in every characteristic.

To prove all the exactness assertions, fix $U_i$ and put $y_j=X_j/X_i$ for $j\ne i$. The Kähler differential sheaf is free there on $d{y}_j$, because a derivation of the polynomial ring is determined freely by its values on the coordinates. Trivialize each $\mathcal{O}(-1)$ with frame $X_i^{-1}$. In this frame,where $e_j$ now denotes the $j$th coordinate vector of the direct sum. The map $\sigma$ is surjective, since its $i$th coefficient is one. Its kernel consists exactly of tuples with $b_i=-{\sum }_{j\ne i}{y}_j{b}_j$, so the $n$ displayed vectors $e_j-y_j{e}_i$ form a free basis of that kernel. The map $D$ sends the differential basis bijectively to this kernel basis and is therefore injective. Exactness on these charts proves the cotangent Euler sequence in homogeneous coordinatesThe proof uses no division by an integer and hence no characteristic-zero assumption.

For $n\ge 1$, the cohomology of twisting sheaves on projective space givesIn particular, $H^0(\mathcal{O})=k$, $H^{q>0}(\mathcal{O})=0$, and $H^q(\mathcal{O}(-1))=0$ for every $q$. The long exact sequence in sheaf cohomology of the displayed Euler sequence beginsand gives zero in the remaining degrees. ThusThe nonzero group is generated by the connecting image of the section $1$ of $\mathcal{O}$. If $n=0$ is admitted, ${\mathbb{P}}^0$ is a point and its cotangent sheaf is zero, so the sole requested group $H^0$ is zero.

Let $u,v,w$ be the consecutive primitive toric ray vectors around the compact invariant curve $F=F_v$. Smoothness gives oriented determinants $\det (u,v)=\det (v,w)=1$, hence $u+w=bv$ for an integer $b$. Choose an algebraic torus character $m$ with $⟨m,u⟩=0$ and $⟨m,v⟩=1$. The principal divisor on a toric variety has coefficient one at $F$, coefficient $b$ at $F_w$, and zero at $F_u$. The other invariant curves do not meet $F$, while each adjacent one meets it transversely once. Intersecting this principal divisor with the proper curve $F$ gives $0=F^2+b$. This proves the self-intersection formula for a toric surface divisor even when the ambient surface is not complete.

If $F^2=-1$, then $b=1$ and $v=u+w$. The vectors $u,w$ form a lattice basis since $\det (u,w)=1$, and their toric cone is the union of the two toric cones adjacent to $v$. Deleting $v$ is consequently a smooth toric fan coarsening. Its inverse is precisely the toric blowup at a torus-fixed point, obtained by inserting the sum of the two primitive basis vectors. Conversely, that insertion gives $u+w=v$, and hence algebraic self-intersection $-1$. Thus the toric contraction of a minus-one curve satisfiesFor the cyclic quotient description, first take a strongly convex toric cone of dimension two. Write its primitive boundary vectors as $a,b$ and put $N_0=\mathbb{Z}a+\mathbb{Z}b$, $r=[N:N_0]$. Since $a$ is primitive, extend it to a basis of $N$ and write $b=ua+rc$ with $\gcd (u,r)=1$. Thus $N/N_0$ is cyclic of order $r$. In the lattice $N_0$, the toric cone is smooth and its toric variety is ${\mathbb{A}}^2$. Passing to the larger lattice $N$ takes the finite quotient whose invariant monomials are precisely those in the smaller dual lattice $M=N^{\ast }\subset N_0^{\ast }$. Relative to the two boundary coordinates, $c=(b-ua)/r$, so a generator acts with weights $(-u,1)$ modulo $r$. Replacing that generator by a suitable power makes the first weight one and the second some $q$ with $0<q<r$ and $\gcd (q,r)=1$. Therefore, for the singular case,The cyclic quotient surface singularity can equivalently be described by the first-quadrant toric cone inIndeed its coordinate ring is $k[x^i{y}^j:i,j\ge 0,i+qj\equiv 0(\mathrm{mod}r)]=k[x,y{]}^{{\mu }_r}$. Characteristic zero ensures the stated ordinary finite-group quotient interpretation.

The restriction to singular toric cones of dimension two is necessary for $r>1$. A smooth toric cone has $r=1$ and gives ${\mathbb{A}}^2$ with the trivial quotient. A toric cone of dimension one gives ${\mathbb{A}}^1\times k^{\ast }$, and the zero toric cone gives $(k^{\ast }{)}^2$. These latter surfaces cannot be a quotient of the displayed kind, since they have nonconstant invertible regular functions, whereas $k[x,y{]}^{{\mu }_r}$ has only constant units. Thus the blanket affine-surface wording requires these separate cases.

For the singular case, compute the negative continued fraction by the ceiling form of the Euclidean algorithm. Start with $a_0=r$, $a_1=q$, and, while $a_i>0$, setThen $0\le a_{i+1}<a_i$, so the process terminates at $a_{s+1}=0$. The gcd is unchanged at each step, hence $a_s=1$. Since $a_{i-1}>a_i>0$, every $b_i\ge 2$. Rearranging the recurrence givesNow define lattice vectorsTo verify this Hirzebruch–Jung resolution algorithm, write $v_i=(p_i,a_i)/r$ with $p_0=0$, $p_1=1$ and $p_{i+1}=b_i{p}_i-p_{i-1}$. Induction gives $a_i\equiv q{p}_i(\mathrm{mod}r)$, so $v_i\in L$. It also gives $p_i{a}_{i+1}-a_i{p}_{i+1}=-r$. Thus every adjacent pair has absolute determinant $1/r$, the covolume of $L$, and is a basis of $L$. The $p_i$ increase strictly, while the $a_i$ decrease to zero, so these toric rays occur in order inside the first quadrant. At termination, $a_s=1$ and the determinant identity gives $p_{s+1}=r$, hence $v_{s+1}=(1,0)$. All boundary and inserted vectors are primitive because they occur in lattice bases.

Subdivide the quadrant by $v_1,\dots ,v_s$. Every resulting toric cone is smooth by the smoothness criterion for a toric variety, and the fan subdivision gives a proper birational toric morphism, hence a resolution. Each interior toric ray has a complete one-dimensional star, so its exceptional curve of a surface resolution is ${\mathbb{P}}^1$. The recurrence yieldsThis proves minimality: no exceptional curve of a surface resolution can be contracted to a smooth point by the Castelnuovo contraction criterion. There is also a direct toric fan check. For fixed $i$, the determinants of $v_i$ with $v_{i+h}$, measured in units of the lattice covolume, satisfy the same recurrence with coefficients at least two and initial values zero and one. They are at least $h$. Omitting any intervening toric ray therefore leaves a toric cone of determinant at least two, so no nontrivial coarsening is smooth. Thus this is the minimal resolution of a cyclic quotient surface singularity.

If all exceptional algebraic self-intersections are $-2$, all $b_i=2$ and induction gives $[2,\dots ,2{]}^{-}=(s+1)/s$. Since $r,q$ are coprime, this forces $r=s+1$, $q=s$. Conversely $r/(r-1)$ has exactly $r-1$ coefficients equal to two. ThereforeFor the final toric fan, the minimal smooth fan subdivision of the toric cone from $(0,1)$ to $(r,1)$ inserts the toric rays $(j,1)$ for $1\le j<r$. Adjacent determinants have absolute value one, and $(j-1,1)+(j+1,1)=2(j,1)$, so its inserted curves have algebraic self-intersection $-2$ and it is the minimal fan subdivision just described. The second original toric cone, from $(r,1)$ to $(1,0)$, is already smooth.

Write $v_j=(j,1)$ and $e=(1,0)$. In the resolved toric fan, the rightmost interior toric ray obeys $v_{r-1}+e=v_r$, so its curve has algebraic self-intersection $-1$ and can be blown down to a smooth point. After deleting it, the same relation $v_{r-2}+e=v_{r-1}$ applies to the new rightmost interior toric ray. Continue deleting $v_r,v_{r-1},\dots ,v_1$ in that order. Each deletion is the inverse of a toric blowup at a torus-fixed point. The last toric fan has only the quadrant toric cone and defines ${\mathbb{A}}^2$. ConsequentlyAll maps induce the identity on the dense algebraic torus, so this composite is the same morphism as $X_{{\Sigma }^{\prime }}\to X_{\Sigma }\to {\mathbb{A}}^2$. The intermediate smooth toric fans need not include the singular toric fan $\Sigma$.

Use the usual irreducible algebraic variety convention here, and write $N=\dim X$, $r=\dim Y$. Fix a closed point $y$ and an affine neighborhood $Y_0$ of it. Noether normalization gives a finite surjective map $\pi :Y_0\to {\mathbb{A}}^r$. Put $a=\pi (y)$. Its fibre is a finite set $\{y_1,\dots ,y_s\}$, one of whose members is $y$, because a finite morphism has finite fibres over closed points.

Choose an affine open $U\subseteq {\varphi }^{-1}(Y_0)$ meeting $X_y$. Since $X$ is irreducible, $U$ still has dimension $N$. In $U$, the fibre of $\pi \varphi$ over $a$ is cut out by the $r$ regular functions ${\pi }_j\varphi -a_j$. Apply the supplied principal hypersurface dimension lemma successively on each irreducible component: if a function vanishes identically the dimension does not drop, if its zero set is empty the component disappears, and otherwise each new component loses exactly one dimension. Hence every surviving component after the $r$ cuts has dimension at least $N-r$.

But that composite fibre is the disjoint union of $U\cap {\varphi }^{-1}(y_i)$. Each member is closed, and since there are finitely many members it is also open in the composite fibre. Thus $U\cap X_y$ is a nonempty union of its irreducible components, givingThe same argument around a point on any fibre component gives the componentwise bound. This is the closed-fibre lower bound by finite normalization; it avoids assuming that $Y$ is smooth or that its point ideal has only $r$ generators.

Irreducibility matters if “variety” is generalized to arbitrary reduced reducible spaces. For example $X={\mathbb{A}}^2\bigsqcup {\mathbb{A}}^1\to {\mathbb{A}}^1$ can map the first component constantly to zero and the second by the identity. It is surjective, but nonzero fibres have dimension zero while $\dim X-\dim Y=1$. For reducible sources one uses the individual component dimensions instead; the preceding proof establishes the stated theorem under its standard irreducible-variety convention.