Abstract
AbstractLet $$X^{1,n}_r$$
X
r
1
,
n
be the blow-up of $$\mathbb {P}^1\times \mathbb {P}^n$$
P
1
×
P
n
in r general points. We describe the Mori cone of $$X^{1,n}_r$$
X
r
1
,
n
for $$r\le n+2$$
r
≤
n
+
2
and for $$r = n+3$$
r
=
n
+
3
when $$n\le 4$$
n
≤
4
. Furthermore, we prove that $$X^{1,n}_{n+1}$$
X
n
+
1
1
,
n
is log Fano and give an explicit presentation for its Cox ring.
Funder
Università degli Studi di Ferrara
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Mathematics
Reference17 articles.
1. Araujo, C., Casagrande, C.: On the Fano variety of linear spaces contained in two odd-dimensional quadrics. Geom. Topol. 21(5), 3009–3045 (2017)
2. Araujo, C., Massarenti, A.: Explicit log Fano structures on blow-ups of projective spaces. Proc. Lond. Math. Soc. 113(4), 445–473 (2016)
3. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23(2), 405–468 (2010)
4. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)
5. Brambilla, M.C., Dumitrescu, O., Postinghel, E., Santana Sánchez, L.J.: Duality and polyhedrality of cones for Mori dream spaces, https://arxiv.org/abs/2305.18536, (2023)