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.

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 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 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.

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.