Degree of a bundle on a generated curve

Let $L$ be ample on $S$ and $s$ sections of $L^d$ with coefficients $v$ whose zero locus is an integral curve $C$ of dimension one. Then $\deg_CL=dL^2$.

Notation. $L^2$ is selfIntersection, namely $\chi(L^2)-2\chi(L)+\chi(\mathcal O_S)$. $\deg_CL$ is curveDegree, namely $\chi(L|_C)-\chi(\mathcal O_C)$. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism. sectionIdeal k s hs v is the comap along sectionsMorphism of the ideal sheaf of the hyperplane with coefficients $v$; sectionCombination k s v is $\sum_j k(v_j)s_j$. IsAmple means that for each point $x$ and open $V\ni x$, some positive power has a global section with nonvanishing open containing $x$, inside $V$, and affine. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$. An IntegralCurve on a surface is an integral closed subscheme of topological Krull dimension one, given by a closed immersion.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.