Some effectivity questions for plane Cremona transformations in the context of symmetric key cryptography

An effective lower bound on the entropy of some explicit quadratic plane Cremona transformations is given. The motivation is that such transformations (Hénon maps, or Feistel ciphers) are used in symmetric key cryptography. Moreover, a hyperbolic plane Cremona transformation g is rigid, in the sense of [5], and under further explicit conditions some power of g is tight.

Identifiability of homogeneous polynomials and Cremona transformations

A homogeneous polynomial of degree d in {n+1} variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in {n+1} variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.

Special cubic Cremona transformations of ℙ6 and ℙ7

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.

Cremona transformations and derived equivalences of K3 surfaces

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.

Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane

As is known, differential geometry studies the properties of curve lines (tangent, curvature, torsion), surfaces (bending, first and second basic quadratic forms) and their families in small, that is, in the neighborhood of the point by means of differential calculus. Algebraic geometry studies properties of algebraic curves, surfaces, and algebraic varieties in general [1; 17]: order, class, genre, existence of singular points and lines, curves and surfaces family intersections (sheaves, bundles, congruences, complexes and their characteristics). Rational curves and surfaces occupy a special place among them: • their design by bi-rational (Cremona) transformations [10; 21]; • investigation of their properties by mapping to lines and planes [9; 21; 22]; • construction of smooth contours from arcs of rational curves belonging to surfaces [10]. It seems that the main results obtained in this direction by mathematicians in the second half of the 19th century by structural and geometric methods should be the theoretical support for the design of technical forms that meet a number of pre-set requirements using modern computational tools and information technologies. It is obvious that application of Cremona transformations’ powerful apparatus is useful when designing, for example, pipes of complex geometry according to set of streamlines, thin-walled shells for a given mesh manifold of curvature lines etc. Apparently, this stage should precede computer graphics’ calculation procedures. However, in Russian publications on applied (engineering) geometry, only a little attention is paid to the study of surfaces in general. The author knows nothing about the use of this approach for solving of these applied problems. In this regard, the aims of this paper are: • illustration of method for mapping a surface to a plane to study its properties in general by the example of construction a flat model for a hyperboloid of one sheet; • constructive approach to the construction of smooth one-dimensional contours on rational surfaces.