Special cubic birational transformations of projective spaces

Abstract

We extend our classification of special Cremona transformations whose base locus has dimension at most three to the case when the target space is replaced by a (locally) factorial complete intersection.

Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds

“…Let Φ : P 6 P 6 be a special cubic Cremona transformation, that is a Cremona transformation defined by cubic forms and whose base locus scheme is smooth and connected. From the main result in [Sta18] (see also [Sta19]), it follows that the base locus of Φ can be of the two following types:…”

“…The surface S 1 satisfies condition K 3 while the surface S 2 does not satisfy this condition and there exist irreducible components of the associated exceptional locus ruled by 6-secant conics. Moreover, from a standard parameter count, one sees that the closure of the locus of smooth cubics containing one of the two types of surfaces S i is C 14 (see [Sta19, Section 4] and Table 3).…”

“…It turns out that R is a linear section of an irreducible fourfold G ⊂ P 7 , singular in dimension 1, and cut out by 7 cubics which define a cubo-cubic Cremona transformation of P 7 . The variety G has the same numerical invariants as the center of a special Cremona transformation of P 7 , whose existence is unknown (see [Sta18] and also [Sta19] for more details on how to extend R). In particular, the general 2-dimensional linear section of the fourfold G ⊂ P 7 is a smooth surface F ⊂ P 5 with the same Hilbert polynomial of R and such that the cubics through the deformations of F describe the divisor C 18 (see [Sta19,Section 4]).…”

“…The variety G has the same numerical invariants as the center of a special Cremona transformation of P 7 , whose existence is unknown (see [Sta18] and also [Sta19] for more details on how to extend R). In particular, the general 2-dimensional linear section of the fourfold G ⊂ P 7 is a smooth surface F ⊂ P 5 with the same Hilbert polynomial of R and such that the cubics through the deformations of F describe the divisor C 18 (see [Sta19,Section 4]). It follows that a cubic fourfold containing R belongs to the intersection…”