By Aleksandr Pukhlikov
Birational tension is a amazing and mysterious phenomenon in higher-dimensional algebraic geometry. It seems that yes common households of algebraic kinds (for instance, three-d quartics) belong to a similar type variety because the projective area yet have notably assorted birational geometric houses. specifically, they admit no non-trivial birational self-maps and can't be fibred into rational kinds via a rational map. The origins of the idea of birational stress are within the paintings of Max Noether and Fano; even though, it was once simply in 1970 that Iskovskikh and Manin proved birational superrigidity of quartic three-folds. This publication provides a scientific exposition of, and a entire advent to, the speculation of birational stress, offering in a uniform approach, rules, suggestions, and effects that to date might in basic terms be present in magazine papers. the new speedy development in birational geometry and the widening interplay with the neighboring parts generate the starting to be curiosity to the rigidity-type difficulties and effects. The e-book brings the reader to the frontline of present study. it really is basically addressed to algebraic geometers, either researchers and graduate scholars, yet can be available for a much wider viewers of mathematicians accustomed to the fundamentals of algebraic geometry
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Birational tension is a notable and mysterious phenomenon in higher-dimensional algebraic geometry. It seems that definite ordinary households of algebraic forms (for instance, 3-dimensional quartics) belong to a similar category style because the projective area yet have significantly diversified birational geometric homes.
Additional info for Birationally Rigid Varieties: Mathematical Analysis and Asymptotics
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.
Birationally Rigid Varieties: Mathematical Analysis and Asymptotics by Aleksandr Pukhlikov