 By B.H. Gross, B. Mazur

Booklet by way of Gross, B.H.

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Additional resources for Arithmetic on Elliptic Curves with Complex Multiplication

Example text

It is easy to select polynomials G and H is such a way that the ﬁbration π : QS → S has a section (and the quartic V is smooth) (examples are given in [IM]). For those special smooth quartics the construction described above gives 24 ALEKSANDR PUKHLIKOV unirationality. Note that the unirationality problem is still open for a generic threedimensional quartic. Therefore, an arbitrary smooth cubic V3 ⊂ P4 and some special smooth quartics uroth problem, however, V4 ⊂ P4 are unirational. The hardest point in solving the L¨ is proving non-rationality of these varieties, which is what was done in the papers of Clemens and Griﬃths [CG] and Iskovskikh and Manin [IM].

In order for the results, obtained in this way, to be meaningful, it is necessary for each set of regularity conditions to show that a variety of general position satisﬁes these conditions; such veriﬁcation is often non-trivial and requires special work. We will explain the methods of checking the regularity conditions. The main result, a complete proof of which is given in Chapter 3, is birational superrigidity (in particular, the absence of non-trivial structures of a rationally connected ﬁbre space, the coincidence of the groups of birational and biregular selfmaps, and, as a very particular corollary, non-rationality) of generic Fano complete intersections Vd1 ·····dk = F1 ∩ · · · ∩ Fk ⊂ PN , deg Fi = di , d1 + · · · + dk = N , of arbitrary dimension (under the restriction k < 12 dim V for the number of the deﬁning equations), and certain other natural classes of Fano varieties.

They show that the assumption of smoothness is essential for the equivalence of rational connectedness and chain rational connectedness. Counting the dimension of the set of rational curves f : P1 → PM of degree d ≥ 1 meeting at dm + 1 points a smooth hypersurface Vm ⊂ PM of degree m, 1 ≤ m ≤ M , it is easy to show that the Fano hypersurface V is rationally connected and, more generally, Fano complete intersections Vm1 ·····mk = Fm1 ∩Fm2 ∩· · ·∩Fmk ⊂ PM , where Fmi is a hypersurface of degree mi , m1 + · · · + mK ≤ M , are rationally connected.