12 introduction

xvi) The projective space of relative dimension n over a scheme X will be denoted

by PX

n

. We omit the subscript when there is no danger of confusion.

xvii) If X is a scheme over a scheme T and Y is a T -scheme, then we also write XY

for the fiber product X ×T Y . If Y = Spec R is aﬃne, then we write XR instead

of XSpec R.

xviii) For X a scheme over a scheme T , we denote by Xt the fiber of X over t ∈ T .

If C is a scheme over the integer ring OKν of the completion Kν of the number

field K with respect to the valuation ν, then we write Cν for the special fiber. In the

particular case that K = and ν = νp, we write Cp instead of Cνp .

If C is a scheme over the integer ring O of a number field K, then we use the same

notation, not mentioning the base change to OKν .

xix) For R any commutative ring, A a commutative R-algebra, and X an R-scheme,

a morphism x: Spec A → X of R-schemes is also called an A-valued point on X.

If A is a field, then we also adopt more conventional language and speak of a point

defined or rational over A. The set of all A-valued points on X will be denoted

by X(A).

xx) If C is a scheme over a valuation ring O and x ∈ C (O), then the reduction of x

is denoted by x.