Examples of base points of linear systems

Question

I'm reading Fulton's algebraic curves book and we have the following definitions:

• A divisor $D=n_1P_1+\ldots,n_kP_k$ ($n_i$'s are integers and $P_i$'s are points) over a curve.
• A linear system as the set of divisors $\{(f)+D\mid f\in L(D)\}$
• A base point of a linear system is a point which is contained in every divisor of the linear system.

I'm a really beginner with this subject which I'm finding very hard. I'm looking for simple examples of base points of a linear system. This would be a great motivation for me to learn these concepts and definitions.

Any help is very welcome!

Thanks

Answer 1

Let $C$ be a smooth projective curve of genus $g$ over an algebraically closed field.
Let $p_1,\cdots, p_n$ be $n$ points on $C$.

Then if you consider the divisor $D=p_1+\cdots+ p_n$, the linear system $\mathcal E\subset |D|$ consisting of the divisors $E\in |D|$ with $p_1$ in their support is tautologically a (non-complete) linear system having $p_1$ as a base point.
If $n\geq 2g$ that linear system $\mathcal E$ is a hyperplane in the projective space $|D|$ of dimension $n-g$ and has thus dimension $n-g-1$ .
Note that $|D|$ itself has no base point if $n\geq2g$.
All this follows from Riemann-Roch but if you haven't learned this theorem yet you can accept the above on faith for the moment if you trussst in meee.

Answer 2

Here is a more advanced example :

Suppose $X$ is a smooth projective curve (over an algebraically closed field) of genus $g\geq 2$ and which is hyperelliptic.
This means that there exists a ramified covering map $p:X\to \mathbb P^1$ of degree 2, which automatically furnishes the involution automorphism $\sigma: X\to X$ interchanging the sheets of $p$ (so that $p(x)=p(\sigma (x))$ for all $x\in X$).
Then for $K$ a canonical divisor the divisor $K-x$ has as base point the point $\sigma (x)$.
To prove this it suffices to prove that $l(K-x)=g-1=l(K-x-\sigma(x))$.
Both equalities follow from Riemann-Roch and and Serre duality.
The second also uses the relation $l(x+\sigma (x))=2$, which follows from the fact that all $p^*(y)\; (y\in \mathbb P^1)$ are linearly equivalent divisors in $X$.