Geodesic
On the rationality of non-singular threefolds with a pencil of Del Pezzo surfaces of degree 4
Sbornik. Mathematics, Tome 197 (2006) no. 1, pp. 127-137 
K. A. Shramov. On the rationality of non-singular threefolds with a pencil of Del Pezzo surfaces of degree 4. Sbornik. Mathematics, Tome 197 (2006) no. 1, pp. 127-137. http://geodesic.mathdoc.fr/item/SM_2006_197_1_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A criterion for the non-singularity of a complete intersection of two fibrewise quadrics in $\mathbb P_{\mathbb P^1}(\mathscr O(d_1)\oplus\dots\oplus\mathscr O(d_5))$ is obtained. The following addition to Alexeev's theorem on the rationality of standard Del Pezzo fibrations of degree 4 over $\mathbb P^1$ is deduced as a consequence: each fibration of this kind with topological Euler characteristic $\chi(X)=-4$ is proved to be rational. Bibliography: 10 titles.

[1] A. V. Pukhlikov, “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii s puchkom poverkhnostei del Petstso”, Izv. RAN. Ser. matem., 62:1 (1998), 123–164 | MR | Zbl

[2] M. M. Grinenko, “Biratsionalnye svoistva puchkov poverkhnostei del Petstso stepenei 1 i 2”, Matem. sb., 191:5 (2000), 17–38 | MR | Zbl

[3] M. Grinenko, “On the birational rigidity of some pencils of del Pezzo surfaces”, J. Math. Sci., 102:2 (2000), 3933–3937 | DOI | MR | Zbl

[4] M. M. Grinenko, “Biratsionalnye svoistva puchkov poverkhnostei del Petstso stepenei 1 i 2. II”, Matem. sb., 194:5 (2003), 31–60 | MR | Zbl

[5] V. V. Przhiyalkovskii, I. A. Cheltsov, K. A. Shramov, “Giperellipticheskie i trigonalnye trekhmernye mnogoobraziya Fano”, Izv. RAN. Ser. matem., 69:2 (2005), 145–204 | MR | Zbl

[6] I. Cheltsov, Nonrational del Pezzo fibrations, , 2004 arXiv:math.AG/0407343

[7] V. A. Alekseev, “Ob usloviyakh ratsionalnosti trekhmernykh mnogoobrazii s puchkom poverkhnostei del Petstso stepeni 4”, Matem. zametki, 41:5 (1987), 724–730 | MR

[8] M. Reid, “Chapters on algebraic surfaces”, Complex algebraic geometry, Lecture notes from a summer program (New York, Park City, Utah, 1993), ed. J. Kollár, Amer. Math. Soc., Providence, RI, 1997, 5–159 | MR | Zbl

[9] G. Brown, A. Corti, F. Zucconi, Birational geometry of 3-fold Mori fibre spaces, , 2004 arXiv: math.AG/0307301 | MR

[10] U. Fulton, Teoriya peresechenii, Mir, M., 1989 | MR