# Quail Entry: Degree |D| = Degree of image

I tried to be careful and detailed in my previous post, but it probably wont happen here.

First goal is to explain how for

(1)

The definition of uses the hilbert function of a projective variety which I wont go into here. Explanation follows most of ex. II.6.2

The setup: , with property that any codim 1 is nonsingular.

### Construct a map. Let be the free abelian group generated by hypersurfaces that doesn’t contain . There is a map

Suffices to define map on , breaks up into irreducible components of codim 1 by

ex I.1.8, ex I.2.6: actually 2.6 is a lengthy problem and the only part of it necessary is that if is projective variety and , then where the are the usual affines of projective space.

so this reduces question to the affine case which ex 1.8 addresses. Replace with . Now = the dimension of a maximal irreducible component, such a component corresponds to a minimal prime .

A comm. alg. result says such minimal primes are ht 1 so the dimension is = . An algebra result says . Result follows.

So the map is where is a local equation for .

The only choice is local equation for , any two choices differ by a unit which doesn’t matter when you take valuations, so map is well defined by extending by linearity.

Principle goes to Principle. Say and none of the contain ; let be the restriction of to . Let be any codim 1 guy in . Compare coeff. of in with .

To accomplish this first describe locally. Use local equations of in a nbd such that . Thus locally . This is valid in a nbd intersecting so both functions agree in nbd of the generic point of therefore

It follows that get map

Connect with Intersection Stuff. With notation above, the intersection of and along , denoted is defined as length over where is the prime corresponding to . This sounds less awkward when everything is stated locally.

Goal is to show if is locally then

. (2)

Make things local: is open affine nbd in of generic point of . The nonsingularity condition says is regular of dimension 1. So in it

Similarly, . Locally can use , so localizing get the module in question is which clearly has length over : .

As above write . Finally, the generalized Bezout Theorem plus (2) gives the latter two equalities in

It follows

Specializing to the case is a curve and is a hyperplane, then is a hyperplane divisor, and so (if is now the divisor of the map to projective space)

which gives (1).

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