Abstract
AbstractIn this paper we study the rationality problem for Fano threefolds $$X\subset {\mathbb P}^{p+1}$$
X
⊂
P
p
+
1
of genus p, that are Gorenstein, with at most canonical singularities. The main results are: (1) a trigonal Fano threefold of genus p is rational as soon as $$p\geqslant 8$$
p
⩾
8
(this result has already been obtained in Przyjalkowski et al. (Izv Math 69(2):365–421, 2005), but we give here an independent proof); (2) a non-trigonal Fano threefold of genus $$p\geqslant 7$$
p
⩾
7
containing a plane is rational; (3) any Fano threefold of genus $$p\geqslant 17$$
p
⩾
17
is rational; (4) a Fano threefold of genus $$p\geqslant 12$$
p
⩾
12
containing an ordinary line $$\ell $$
ℓ
in its smooth locus is rational.
Funder
Università degli Studi di Roma Tor Vergata
Publisher
Springer Science and Business Media LLC
Reference22 articles.
1. Beauville, A.: Variétés de Prym et jacobiennes intermédiaires. Annales scientifiques de l’É.N.S. 4e série 10(3), 309–391 (1977)
2. Cheltsov, I.: Nonrational nodal quartic threefolds. Pac. J. Math. 226(1), 65–81 (2006)
3. Chiantini, L., Ciliberto, C.: Threefolds with degenerate secant varieties: on a theorem of G. Scorza, Marcel Dekker L.N. 217, 111–124 (2001)
4. Ciliberto, C., Fontanari, C.: On varieties whose general surface section has negative Kodaira dimension, arXiv:2305.08730
5. Conte, A., Murre, J.P.: On the definition and on the nature of the singularities of Fano threefolds. Rend. Sem. Mat. Univ. Politec. Torino 44, 51–67 (1986)