Aven Bross

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Very ample divisors on curves

Dec 11, 2018

All of the sources that I have found characterizing very ample divisors reference Hartshorne or otherwise use, in my opinion, “non-elementary” arguments. Here are some “elementary” proofs that I worked out with some fellow grad students.

Let be a projective non-singular irreducible curve over an algebraically closed field. Suppose that is a divisor on such that Let

Here $L(D)$ is the Riemann-Roch space of $D$ and $l(D)$ is the dimension of $L(D)$. We define the map by

Lemma 1. If and are linearly equivalent divisors, then there is an automorphism of given by linear forms such that

Let be such that Then the map gives an isomorphism

If is defined by the basis , then

By the isomorphism above, given any basis for the set is a basis for Thus there exists a change of basis on defined by linear forms such that for each Therefore also gives a projective linear map defined by

Moreover, as desired.

Definition 2. A divisor $D$ is said to be very ample if the map gives an isomorphism to its image.

Theorem 3. A divisor with is very ample if and only if for every two points we have

Suppose that is very ample. Let Pick such that Observe that by applying a linear automorphism to (changing basis for ) we can assume that

Let be defined by the basis Then and Since is effective, we must have and Moreover, and Thus similarly we have but So

It remains to show that

Note we have already shown above that

Suppose that

Then every that has a zero at must have a double zero at Note that must be a local parameter at on for some $i\in{2,\ldots,n}$ since generate But by our assumptions above it must be that

This is a contradiction to being an isomorphism.

Conversely suppose that

for all We will first show that is injective. Let and assume By our assumption we may pick

and therefore and Thus

It remains to show that is nonsingular. Again suppose that Let us pick

and

We may then extend this to a basis and note that

Note that we may find a non-singular model for Therefore there exists a birational map between and and hence a is a birational map from to So is an isomorphism.

Observe that generates Moreover, for we have

and

Therefore for $i\ge 3$ we have

where and $u$ is a unit in So and thus has order $m$ at $\phi_D(P)$. Thus is a basis for and is nonsingular at Thus is nonsingular and defines an isomorphism.