The cubo-cubic transformation and K3 surfaces

A rough classification of potentially invertible cubic transformations of the real plane

A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called cubic if the degrees of its polynomials are not greater than three. In the present paper a rough classification scheme for cubic transformations of $\Bbb R^2$ is suggested. It is based on quartic forms associated with these transformations.

On quartic forms associated with cubic transformations of the real plane

A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called cubic if the degrees of its polynomials are not greater than three. It turns out that cubic transformations are associated with some binary and quaternary quartic forms. In the present paper these forms are defined and studied.

Special cubic birational transformations of projective spaces

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.

由Cremona变换产生的新有理三次四维流形

New rational cubic fourfolds arising from Cremona transformations

本文证明判别式20的非常一般三次四维流形中Fourier-Mukai等价蕴含双有理等价，并借助Veronese曲面定义的Cremona变换发现新的有理三次四维流形。

通过Enriques Cremona变换的双有理三次四fold

Birational cubic fourfolds via an Enriques Cremona transformation

本文通过构造一个由Enriques曲面Fano模型上二重五次线性系统给出的Cremona变换，建立了判别式44的非常一般三次四fold与其唯一Fourier–Mukai伴侣之间的双有理等价。

Cremona transformations, surface automorphisms and plane cubics

We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure on the cubic to understand when the indeterminacy and exceptional behavior of the transformation may be eliminated by repeated blowing up.

Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric …

Automorphisms of cubic surfaces without points

We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $K$ of characteristic zero that has no $K$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

Quotients of groups of birational transformations of cubic Del Pezzo fibrations

We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.

The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces

Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for stable cubic surfaces: the moduli space is biholomorphic to a quotient of the compex 4-ball by an explict arithmetic group generated by complex reflections. This identification gives interesting structural information on the moduli space and allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Ferm…

四次曲面网络与平面三次曲线及其GIT稳定性

Nets of quadric surfaces and plane cubics and their GIT stability

本文研究四次曲面网络与平面三次曲线的对应关系，通过Gale变换定义，并利用birational-geometric技术提供三重几何对象的GIT稳定性判据。

New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves

For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil.

Composition of points and Mordell-Weil problem for cubic surfaces

Let $V$ be a plane smooth cubic curve over a finitely generated field $k.$ The Mordell-Weil theorem for $V$ states that there is a finite subset $P\subset V(k)$ such that the whole $V(k)$ can be obtained from $P$ by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with $V.$ Equivalently, the group of birational transformations of $V$ generated by reflections with respect to $k$-points is finitely generated. In this paper, elaborating an idea from [M3], we establish a Mordell-Weil type finite generation result for…

A geometric framework for the subfield problem of generic polynomials via Tschirnhausen transformation

Let $k$ be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over $k$ via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for $\frak{S}_3$ and $C_3$ over $k$. As an application of the cubic case, we also give several sextic generic polynomials over $k$.

Twistor lines on cubic surfaces

It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.