 Residue trivialization for a degree n+1 projective hypersurface

ID: residue-trivialization-for-a-degree-n-plus-1-projective-hypersurface

Y = {f = 0} \subset CP^{n}

of degree

n + 1

, let

E = \sum z_{i} \partial_{z_{i}}

and

Ω = d z_{0} \land \dots \land d z_{n}

. The meromorphic form

(ι_{E} Ω) / f

descends to projective space because its scaling weights cancel and it is horizontal. Its Poincaré residue is a nowhere-zero section of

K_{Y}

. On the chart

z_{j} = 1

, with remaining coordinates

t_{1}, \dots, t_{n}

in original-index order and

F = f ∣_{z_{j} = 1}

, it is

(- 1)^{j + r - 1} (d t_{1} \land \dots d t_{r} \dots \land d t_{n}) / F_{t_{r}}

wherever

F_{t_{r}} \neq = 0

. Thus

K_{Y} ≅ O_{Y}

 New to topics? Read the docs here!