Degree of the total exceptional ideal

Let $B$ be a point blow-up with an ample bundle on its surface, $J,\iota$ presenting the pullback of the centre ideal, and $C$ an integral curve on the blow-up. Then $\deg_CJ=\sum_i\deg_CE_i$.

Notation. $\deg_CL$ is curveDegree, namely $\chi(L|_C)-\chi(\mathcal O_C)$. PresentsPullbackIdeal I f J ι says that $\iota:J\to\mathcal O_Y$ is a monomorphism whose image on affine opens is the extension of $I$ along $f$. 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. Configuration S r is the set of $r$-tuples of complex points with distinct images, with the Zariski topology induced from the product.

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.