Rational functions, divisors and differentials

For a smooth projective curve over an algebraically closed field, a nonzero rational function has an integer order at each point: positive for a zero, negative for a pole, and zero for a local unit. Its divisor is the finite sum of these orders times the corresponding points. Principal divisors have total degree zero. Over a nonclosed field, closed points carry residue-field degrees; working geometrically first makes the multiplicities in the following examples especially transparent.

For a short Weierstrass equation in characteristic zero, a local parameter at O is t=−x/y. Its leading expansions are x=t to the minus two plus higher terms and y=−t to the minus three plus higher terms. Thus x has a double pole at O and y a triple pole. At a branch point with y=0, however, y itself can be a local parameter, and the order of x minus its branch value is two. Counting only distinct roots loses precisely this ramification information.

Let Pic-zero denote degree-zero divisors modulo principal divisors. For a genus-one curve with O, the map P to the class of P minus O is bijective over an algebraic closure. Indeed, adding O to a degree-zero divisor class produces a degree-one class. Riemann–Roch gives a nonzero section of that class, hence an effective divisor of degree one, which is one point. Uniqueness follows because a nonconstant function with only one simple pole would give an impossible degree-one map to the projective line.

Transporting addition in Pic-zero supplies an associative operation on points. A nonvertical affine line has three zeros on the cubic, counted with multiplicity, and a triple pole at O. Its principal divisor therefore shows that the three intersection classes sum to zero. This recovers the chord–tangent law. The reasoning takes place on a smooth projective curve, where local orders and Riemann–Roch apply; copying it onto a singular plane equation without normalization would change the hypotheses.

For a smooth short model in characteristic zero, omega=dx/(2y) extends to a nowhere-vanishing regular differential. Where y vanishes, the other denominator is nonzero by smoothness, so the expression in dy is regular. At O, substituting the leading t-expansions gives omega=(1 plus higher-order terms) dt. It is therefore regular and nonzero there as well. The space of global regular differentials has dimension one, and translation by any point preserves this invariant differential.

Riemann–Roch also explains the basic Weierstrass coordinates: functions with pole bounded by two O form a two-dimensional space spanned by one and x, while the bound three O adds y. These pole budgets organize later constructions of maps and division functions. A rational function may have balanced zeros and poles without being constant; only an everywhere regular function on a projective integral curve is constant. Keeping those statements separate prevents the degree-zero condition from being mistaken for a vanishing divisor.