Geodesic
Congruences of conics in $\mathbf{P}^3$
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (1982), pp. 57-62 
V. A. Iskovskikh. Congruences of conics in $\mathbf{P}^3$. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (1982), pp. 57-62. http://geodesic.mathdoc.fr/item/VMUMM_1982_6_a11/
@article{VMUMM_1982_6_a11,
     author = {V. A. Iskovskikh},
     title = {Congruences of conics in $\mathbf{P}^3$},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {57--62},
     year = {1982},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1982_6_a11/}
}
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Let $\pi\colon V\to S$ be a conic bundle with a discriminant locus $C\subset S$. We claim the following rationality criterion. $V$ is rational if there exists a pencil of rational curves $\{L_\lambda\subset S|_\lambda\in \mathbf{P}^1\}$ such that either $(L_\lambda\cdot C)\le3$ or $S=\mathbf{P}^3$, $\deg C=5$ and $\pi$ corresponds to an even $\theta$-characteristic. Here we prove the “only if” part of the criterion. The “if” part is reduced to a question of birational equivalence of the congruence of rational curves in $\mathbf{P}^3$ to the congruence of conies.