Morphism of algebraic varieties

A morphism of algebraic varieties is a map given locally by regular functions. A projective-coordinate formula defines a morphism wherever its homogeneous coordinate functions do not vanish simultaneously.

A morphism of algebraic varieties whose source and target are algebraic curves. A nonconstant morphism between smooth projective curves is finite and surjective; its degree is the degree of the induced extension of function fields.

A flat proper morphism of algebraic varieties whose geometric fibres are plane conics, usually presented as a quadratic equation in a projective bundle of relative dimension two. The generic fibre is normally required to be smooth; singular conics can occur in special fibres. A conic bundle from a line on a smooth cubic surface has five reducible fibres over the projective line.

In characteristic different from two, a local equation for a conic bundle is $z^{\mathsf{T}}Mz=0$ with $M$ symmetric. A fibre is singular exactly when the determinant vanishes. Changing the fibre coordinates multiplies that determinant by a nonzero square, so its zero locus is intrinsically defined even when the quadratic coefficients belong to twisting line bundles.

Let the base be a smooth algebraic curve, the total surface be smooth, and the field be algebraically closed of characteristic different from two. At a rank-two singular fibre with kernel vector $v$, the fibre derivatives vanish at $[v]$. Smoothness requires $v^{\mathsf{T}}{M}^{\prime }v\ne 0$. The adjugate matrix is a nonzero multiple of $v{v}^{\mathsf{T}}$, so the determinant has a simple zero. At rank at most one, the projective kernel contains a line and the quadratic $v^{\mathsf{T}}{M}^{\prime }v$ has a zero on it; that point would make the total surface singular. Thus every singular fibre is a pair of distinct projective lines and its discriminant zero is simple.

A dominant morphism of varieties is generically finite when its extension of function fields is finite. Its degree is the degree of this field extension. A dominant morphism between integral varieties of equal dimension is generically finite; over a suitable nonempty open subset of the target its fibers are finite.

A $k$-algebra isomorphism between the local rings at closed points extends to a morphism on some neighborhoods: represent the images of finitely many affine coordinate generators by fractions, and shrink where their denominators are nonzero and their finitely many relations hold. Extend its inverse the same way. The two compositions agree with the identity as germs; finitely many coordinate equalities make them identities after shrinking again. Taking inverse images of these neighborhoods then gives mutually inverse actual neighborhood isomorphisms, not just a formal-completion isomorphism.

A nonconstant morphism of algebraic varieties $f:C\to D$ between smooth projective integral curves is finite, and its degree is the degree of the induced extension of function fields. It is also the degree of the pullback of a rational point, with ramification multiplicities and residue degrees included: locally the finite coordinate algebra is a torsion-free module over the target discrete valuation ring, hence free of its generic rank. For $D={\mathbb{P}}^1$ and a rational function defining $f$, the pullback of infinity is its pole divisor, so the total pole degree equals $\mathrm{deg}f$. A degree-one finite map to a normal curve is an isomorphism, since a finite integral algebra inside its unchanged function field equals the integrally closed target ring.

For an affine variety with coordinate ring $A$ acted on by a finite group $G$, the affine quotient is $\mathrm{Spec}{A}^G$. Elements of $A$ satisfy monic orbit polynomials over $A^G$, so the quotient map is a finite morphism. Over an algebraically closed field, invariant functions separate finite orbits; in characteristic not dividing $|G|$, interpolate and average a function separating two orbits. The quotient fibres are consequently the orbits.

For a classical algebraic variety $X$ and affine variety $Y$, a homomorphism on the indicated rings gives a morphism by evaluating the images of affine coordinate generators at each point of $X$. The relations defining $Y$ remain zero. Local fractions pull back to local regular fractions, proving regularity; the coordinate ring generators also prove uniqueness.

For a nonconstant morphism between nonsingular curves and a local target uniformizer $t$ at the image of $P$, the index is the order of vanishing of its pullback. Changing $t$ multiplies it by a local unit in a ring and does not change that order. This is a valuation definition and remains meaningful for inseparable morphisms.

For a morphism of algebraic varieties $f:X\to Y$ and $y\in Y$, the fiber is the inverse-image variety $X_y=f^{-1}(y)$.

For a point $y$, the underlying fiber is the inverse image $f^{-1}(y)$.

The dimension of the fibre of a morphism locally at its source point $x$: the maximum of the dimensions of the fibre's irreducible components through $x$. This differs from the Krull dimension of the local ring at a nonclosed point.

For a finite-type morphism of schemes, local fibre dimension is upper semicontinuous on the source, in the displayed sense. If the morphism is proper, the maximal fibre dimension is also upper semicontinuous on the target. Without properness a component of a large fibre can disappear from the source over a special point, so the target statement is not valid in that generality. The Stacks Project: dimension of fibres distinguishes these statements.

For a dominant morphism of irreducible varieties, generic fiber dimension is source dimension minus target dimension, and this dimension occurs over a nonempty open subset. In particular, constant fiber dimension $d$ gives source dimension equal to target dimension plus $d$. For algebraic groups the equal-dimensional components and coset fibers allow the same dimension formula componentwise.

For an irreducible source, regard the morphism of algebraic varieties as dominant onto the closure of its image, with generic point $\eta$. The fiber dimension theorem gives the displayed formula. A proper morphism has closed image, so the closure can then be omitted.

Let $f:X\to Y$ be a proper morphism of irreducible algebraic varieties with equal dimension. If some fibre has an isolated point, its corresponding irreducible component has dimension zero. Apply the fiber dimension theorem to $X\to f(X)$: $0\ge \dim X-\dim f(X)$. Thus the closed image has dimension $\dim Y$ and equals $Y$. The incidence proof that a cubic surface contains a line is an application; the entire witness fibre need not be finite.

For a dominant morphism of algebraic varieties between irreducible varieties, there is a nonempty Zariski-open subset of the target over which every fibre is nonempty and every irreducible component has the displayed dimension. Apply Noether normalization to affine source charts over the target function field and clear denominators on the target to get a generic upper bound. Combine it with the closed-fibre lower bound in the fiber dimension theorem.

For a surjective morphism of irreducible varieties, choose an affine neighborhood of the target point and a finite Noether normalization to affine space of target dimension $r$. On an affine source neighborhood the composite fibre is cut out by $r$ functions, so repeated principal hypersurface dimension lemma gives dimension at least source dimension minus $r$ on every component. The normalization fibre is finite. The original fibre is therefore an open-and-closed union of components of that composite fibre, proving the bound without assuming target smoothness. For a reducible source the bound must instead be applied using the dimension of each source component.

For a morphism of schemes $f:X\to Y$ and a point $y\in Y$, the scheme-theoretic fiber is the fiber product $X{\times }_Y\mathrm{Spec}\kappa (y)$. It remembers multiplicities and nilpotents that the underlying set-theoretic inverse image cannot detect.

Its underlying fiber is enriched with local-ring and multiplicity information.

Over a perfect field $k$ of characteristic $p$, the scheme-theoretic fiber of the Absolute Frobenius morphism of ${\mathbb{A}}_k^1$ over a rational point $a$ isIts underlying space is one point, but its coordinate ring has a nonzero nilpotent element, so the fibre is a length-$p$ nonreduced scheme.

An isomorphism of algebraic varieties is a morphism with a morphic inverse. It preserves all properties intrinsic to the variety.

A finite morphism is a morphism whose inverse image of every affine open set is affine and whose corresponding coordinate ring is finite as a module over the base coordinate ring. A nonconstant morphism between smooth projective curves is finite, and its degree is the number of points in a generic fiber counted with multiplicity.

For a finite generically separable morphism between smooth projective curves, the field trace on rational differentials identifies the differential sheaf with the trace-dual module. Locally, for $A=k[[s]]\subset B=k[[u]]$ with $s=f(u)$, the trace-dual fractional ideal is $f^{\prime }(u{)}^{-1}B$. Multiplying it by $ds=f^{\prime }(u)du$ gives $Bdu$. For a monogenic finite extension with monic polynomial $P$, the identity $\mathrm{Tr}(g(u)/P^{\prime }(u))=[T^{e-1}](g(T)\mathrm{mod}P(T))$ proves the dual-module formula: the top-coefficient pairing on the basis $1,u,\dots ,u^{e-1}$ has invertible antitriangular matrix. Tensoring locally with an invertible sheaf yields ${\pi }_{\ast }({\mathcal{M}}^{\vee }\otimes {\Omega }_C^1)\cong \mathcal{H}om({\pi }_{\ast }\mathcal{M},{\Omega }_D^1)$.

A finite morphism is of finite type and separated: on affine charts the multiplication map $B{\otimes }_{A}B\to B$ is surjective and represents its closed diagonal. Each element of a finite algebra satisfies a monic equation by the determinant trick. In a valuative lifting problem, its image in the fraction field is therefore integral over the valuation ring and belongs to that ring, since valuation rings are integrally closed. This gives a lift, unique because the valuation ring embeds in its fraction field. The valuative criterion for properness proves properness.