# Sheaves

\gdef\mc#1{\mathcal{#1}} \gdef\mf#1{\mathfrak{#1}}

# Invertible Sheaves or Line Bundles

# The Tautological and Hyperplane Bundles

Consider the line bundle

OPn(1):=(Cn+1{0})×CC=((Cn+1{0})×C)/C, \mc{O}_{\PP^{n}}(1) := \big( \CC^{n+1} - \{0\} \big) \times_{\CC^{\ast}} \CC = \bigg( \big( \CC^{n+1} - \{0\} \big) \times \CC \bigg) / \CC^{\ast},

where C\CC^{\ast} acts via inverse scalar multiplication on both Cn+1\CC^{n+1} and C\CC, i.e. if Cn+1=Spec(C[x0,,xn])\CC^{n+1} = \Spec(\CC[x_{0},\ldots, x_{n}]) and C=Spec(C[y])\CC = \Spec(\CC[y]), then each variable has weight one with respect to the C\CC^{\ast}-action.

OPn(1)\mc{O}_{\PP^{n}}(1) is a space that maps to Pn\PP^{n} by projecting onto the first factor, and the fibre of this projection over any point is a line. More generally, we can consider the line bundle

OPn(d):=(Cn+1{0})×CCd=((Cn+1{0})×Cd)/C, \mc{O}_{\PP^{n}}(d) := \big( \CC^{n+1} - \{0\} \big) \times_{\CC^{\ast}} \CC_{d} = \bigg( \big( \CC^{n+1} - \{0\} \big) \times \CC_{d} \bigg) / \CC^{\ast},

where now C\CC^{\ast} acts on Cd=Spec(C[y])\CC_{d} = \Spec(\CC[y]) with weight dd, i.e. ty=tdyt \cdot y = t^{d}y for any tCt \in \CC^{\ast} and with dZd \in \ZZ.

::: proposition Tautological and Hyperplane Bundles The set OPn(1)Pn×Cn+1\mc{O}_{\PP^{n}}(-1) \subset \PP^{n} \times \CC^{n+1} that consists of all pairs (l,z)Pn×Cn+1(l,z) \in \PP^{n} \times \CC^{n+1} with zlz \in l forms in a natural way a holomorphic line bundle OPn(1)\mc{O}_{\PP^{n}}(-1) over Pn\PP^{n}, called the tautological bundle.

Its dual bundle OPn(1):=OPn(1)\mc{O}_{\PP^{n}}(1) := \mc{O}_{\PP^{n}}(-1)^{\ast} is called the hyperplane or anti-tautological bundle. :::