Studying algebraic geometry, while the abstract theory is pretty clear to me I often feel puzzled in practice. Here I am trying to understand linear equivalence of divisors in some practical situations. Consider the following example.
Let $S\subset\Bbb{P}^3$ be a smooth surface of degree $d\geq3$ and let $\ell\subset S$ be a line. A plane containing $\ell$ cuts on $S$ the divisor
$$ H_0=\ell+C$$
where $C\subset S$ is a curve of degree $d-1$. Let $H$ be the divisor cut on $S$ by a generic plane. So $H$ cuts $\ell$ in one point. I am not sure about the following:
Question: why can we say that $H_0\sim H \ $ (linear equivalence) ?
This is then a nice example for showing that even the most simple divisor like $\ell$ can have negative self-intersection: if $H_0\sim H$ then
$$1=H\cdot\ell=H_0\cdot\ell=(\ell+C)\cdot\ell=\ell^2+C\cdot\ell=\ell^2+d-1 $$
Hence $\ell^2=2-d<0$.
Best Answer
(Turning my comment into an answer...) The point is that restriction of divisors respects linear equivalence (as follows quickly from the definition). Any two planes in the ambient space $\mathbf{P}^3$ are linearly equivalent divisors (easy exercise!), so the same remains true when we restrict them to the surface $S$.