scholarly journals Cremona transformations and derived equivalences of K3 surfaces

2018 ◽  
Vol 154 (7) ◽  
pp. 1508-1533 ◽  
Author(s):  
Brendan Hassett ◽  
Kuan-Wen Lai
Keyword(s):  
K3 Surfaces ◽  
Cremona Transformation ◽  
Grothendieck Ring ◽  
Cremona Transformations ◽  
Derived Equivalences ◽  
Affine Line ◽  
The Difference ◽  
Base Loci

We exhibit a Cremona transformation of $\mathbb{P}^{4}$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

scholarly journals Autoequivalences of twisted K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 912-937 ◽  
Author(s):  
Emanuel Reinecke
Keyword(s):  
K3 Surfaces ◽  
K3 Surface ◽  
Lattice Structures ◽  
Hodge Structures ◽  
Derived Equivalences

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index $1$or $2$in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.

scholarly journals On the non-divisorial base locus of big and nef line bundles on K3[2]-type varieties

2020 ◽  
pp. 1-27
Author(s):  
Ulrike Rieß
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Ample Line Bundle ◽  
Base Point ◽  
K3 Surfaces ◽  
Line Bundles ◽  
Codimension Two ◽  
Base Locus ◽  
Base Loci ◽  
Symplectic Varieties

Abstract We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

scholarly journals Special cubic Cremona transformations of ℙ6 and ℙ7

2019 ◽  
Vol 19 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Giovanni Staglianò
Keyword(s):  
Cremona Transformation ◽  
Cremona Transformations ◽  
Base Locus ◽  
Famous Result

Abstract A famous result of Crauder and Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. They also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of ℙ5; 2) a cubo-quintic transformation of ℙ6; or 3) a quadro-quintic transformation of ℙ8. Special Cremona transformations as in Case 1) have been classified by Ein and Shepherd-Barron (1989), while in our previous work (2013), we classified special quadro-quintic Cremona transformations of ℙ8. Here we consider the problem of classifying special cubo-quintic Cremona transformations of ℙ6, concluding the classification of special Cremona transformations whose base locus has dimension three.

scholarly journals The class of the affine line is a zero divisor in the Grothendieck ring: Via $G _{ 2 }$-Grassmannians

2018 ◽  
Vol 28 (2) ◽  
pp. 245-250 ◽  
Author(s):  
Atsushi Ito ◽  
Makoto Miura ◽  
Shinnosuke Okawa ◽  
Kazushi Ueda
Keyword(s):  
Zero Divisor ◽  
Grothendieck Ring ◽  
Affine Line

The resolution of monoidal Cremona transformations of three-dimensional space

1938 ◽  
Vol 34 (1) ◽  
pp. 22-26
Author(s):  
D. W. Babbage
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Suitable Choice ◽  
Cremona Transformation ◽  
Cremona Transformations ◽  
Image Position ◽  
Three Dimensional Space

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the formwhere φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).

scholarly journals Derived equivalences of K3 surfaces and orientation

2009 ◽  
Vol 149 (3) ◽  
pp. 461-507 ◽  
Author(s):  
Daniel Huybrechts ◽  
Emanuele Macrì ◽  
Paolo Stellari
Keyword(s):  
K3 Surfaces ◽  
Derived Equivalences

scholarly journals On the Stereometric Generation of the De Jonquières Transformation

1918 ◽  
Vol 37 ◽  
pp. 48-58
Author(s):  
J. F. Tinto
Keyword(s):  
Three Dimensions ◽  
Plane Figure ◽  
Cremona Transformation ◽  
Quartic Surface ◽  
Cremona Transformations ◽  
Intrinsic Interest ◽  
Four Dimensions ◽  
Single Instance

In the geometry of the plane the logical interrelations of figures may often be rendered clearer by considering the plane to be a part of space of three dimensions. Thus, by taking the plane figure as part of a more extensive configuration in space of three dimensions, the elucidation of its properties, and in particular its relation with other figures, are often facilitated. Similarly, the figures of space of three dimensions can sometimes be treated more advantageously and compendiously by considering them as parts of figures in a space of four dimensions, and so on. As a single instance we may take Segre's elegant and powerful mode of treatment of the quartic surface which possesses a nodal conic. This surface he obtains as a projection in space of four dimensions of the quartic surface which constitutes the base of a pencil of quadratic varieties. In the following paper this mode of treatment has been applied to the interesting variety of the Cremona transformation in the plane known as the De Jonquieres transformation, a transformation which possesses some intrinsic interest in view of the fundamental rôle which it plays in the theory of Cremona Transformations. By the aid of a surface in space of three dimensions, a variety in space of four dimensions, etc., simple constructions are given for the De Jonquières transformation between two planes, between two spaces of three dimensions, etc., respectively.

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