In this paper we investigate the divisor C14 inside the moduli space of smooth cubic hypersurfaces in P 5 , whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in P 5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in C14 is rational. Combining our proof with the Hodge theoretic definition of C14, we deduce that on a smooth cubic fourfold every class T ∈ H 2,2 (X, Z) with T 2 = 10 and T · h 2 = 4 is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in C14.
The works of Hassett and Kuznetsov identify countably many divisors C d in the open subset of P 55 = P(H 0 (O P 5 (3))) parametrizing all cubic 4-folds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family C14. We use congruences of 5-secant conics to prove rationality for the first three of the families C d , corresponding to d = 14, 26, 38 in Hassett's notation.
We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.Keywords birational transformation · base locus · quadric hypersurface · entry locus · tangential projection of π P n , i.e. as Sec(X) := π P n (S X ) =
We construct new examples of rational Gushel-Mukai fourfolds, giving more evidence for the analogous of the Kuznetsov's Conjecture for cubic fourfolds: a Gushel-Mukai fourfold is rational if and only if it admits an associated K3 surface.
The Macaulay2 package Cremona performs some computations on rational and birational maps between irreducible projective varieties. For instance, it provides methods to compute degrees and projective degrees of rational maps without any theoretical limitation, from which is derived a general method to compute the push-forward to projective space of Segre classes. Moreover, the computations can be done both deterministically and probabilistically. We give here a brief description of the methods and algorithms implemented.
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