Existence of the point blow-up structure

For a surface $S$ with an ample bundle $L$ and any configuration $p$ of $r$ points, a PointBlowup S r p exists: a surface with a blow-up of the centre ideal over $\mathbb C$, together with exceptional line bundles presenting the pullbacks of the ideals of each point.

Notation. 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 $\mathbb C$. 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.