Rend. Circ. Mat. Palermo, II. Ser (2017) 66:205–216
Cones of effective divisors on the blown-up
· Elisa Postinghel
Received: 2 January 2016 / Accepted: 22 October 2016 / Published online: 1 December 2016
© Springer-Verlag Italia 2016
Abstract We compute the facets of the effective cones of divisors on the blow-up of
up to ﬁve lines in general position. We prove that up to six lines these threefolds are weak
Fano and hence Mori Dream Spaces.
Keywords Linear series · Effective cone · Fano threefolds · Mori dream spaces · Fat lines
Mathematics Subject Classiﬁcation Primary: 14C20; Secondary: 14J70, 14J26
In classical algebraic geometry, the study of linear systems in
of hypersurfaces of degree
d with prescribed multiplicities at a collection of s points in general position was investigated
by many authors for over a century (see  for an overview).
We will brieﬂy recall the most important results in this area. The well-known Alexander–
Hirschowitz theorem  classiﬁes completely the dimensionality problem for linear systems
with double points (see [5,7,8,19] for more recent and simpliﬁed proofs). Besides this the-
The ﬁrst author is a member of the Simion Stoilow Institute of Mathematics of the Romanian Academy. The
second author is supported by the Research Foundation—Flanders (FWO). The third author is supported by
the PISCOPIA cofund Marie Curie Fellowship Programme.
Max-Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK
Dipartimento di Matematica, Universita’ degli Studi di Padova, Via Trieste 63, 35121 Padua, Italy