manuscripta math. 98, 155 – 163 (1999) © Springer-Verlag 1999
Cícero F. Carvalho
Linear systems on singular curves
Received: 15 June 1998
Abstract. We start this work by studying free linear systems on singular curves and related
base point free linear systems on the non-singular model. We apply these results to the study
of pencils of small degree on non-singular curves. We also prove a “base point free pencil
trick” which holds for any (possibly) singular curve.
1. Introduction
We begin by reviewing in this section the theory of divisors and linear
systems on Gorenstein curves as presented in [7] and [2].
Let X be a complete integral curve defined over an algebraically closed
field k of characteristic greater than or equal to zero and let k(X) be its
function field. A divisor D on X is a non-zero coherent fractional ideal
sheaf of X, which we denote by the product of its stalks D =
P ∈X
D
P
. The
product and quotient of two divisors D and E is defined by D· E =
D
P
E
P
and D : E =
(D
P
: E
P
) =
{h ∈ k(X)|hE
P
⊆ D
P
}. We will write
D ⊇ E when D
P
⊇ E
P
for all P ∈ X, note that this defines an order in the
set of divisors of X.
The structure sheaf O of X is trivially a divisor O =
O
P
. For h ∈
k(X)
∗
we define div h :=
(1/h)O
P
. The k-vector space of global sections
of D is H
0
(D) ={h ∈ k(X)
∗
|(div h)
P
D
P
⊇ O
P
∀ P ∈ X}∪{0}=
P ∈X
D
P
. The local degree of a divisor at P ∈ X is defined by the properties
deg
P
(O
P
) = 0 and deg
P
(D
P
) − deg
P
(E
P
) = dim
k
(D
P
/E
P
) if D
P
⊇ E
P
.
The degree of a divisor D on X is deg
X
(D) =
∑
P ∈X
deg
P
(D
P
) and is
finite because we have D
P
= O
P
almost always. Most of the time we write
simply deg D.
Let V ⊆ H
0
(D) be a k-vector subspace of dimension r + 1, then V
induces a linear system on X of dimension r and degree d = deg D which
is defined by g
r
d
={(div s) · D | s ∈ V \{0}}. We say that g
r
d
is base point
free if for each P ∈ X there exists s
P
∈ V \{0} such that (1/s
P
)D
P
= O
P
.
C. F. Carvalho: Universidade Federal de Uberlândia, DEMAT,Av. Universitaria s/n.,
38400 Uberlândia - MG, Brazil. e-mail: cicero@impa.br
Mathematics Subject Classification (1991): 14C20, 14C21