Sonia Brivio, Gian Pietro Pirola
Let X be a smooth complex projective surface. We prove that for any sufficiently big m there exists a rational dominant map f from X into a complex rational ruled surface Y, such that f is generically finite of degree m and has monodromy the alternating group Am.
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Let $X$ be a generic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p\geq 0$. We show that for $n$ sufficiently large there exists a tame rational map $f:X\to \PP^1_k$ with monodromy group $A_n$. This generalizes a result of Magaard--Völklein to positive characteristic.
We shall determine the uniquely existing extension of the alternating group of degree 6 (being normal in the group) by a cyclic group of order 4, which can act on a complex K3 surface.
In this note, we consider all possible extensions G of a non-trivial perfect group H acting faithfully on a K3 surface X. The pair (X, G) is proved to be uniquely determined by G if the transcendental value of G is maximum. In particular, we have G/H < Z/(2) + Z/(2), if H is the alternating group A_5 and normal in G.
It is proved that there exist plane rational curves of degree twelve (resp. twenty-four) with two different outer Galois points such that the Galois group at one of two Galois points is an alternating group $A_4$ (resp. a symmetric group $S_4$) of degree four, under the assumption that the characteristic of the ground field is eleven (resp. is twenty-three). For an alternating group $A_5$ of degree five, a similar existence theorem is confirmed, over a field of characteristic $59$, by GAP system.
Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over projective line by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group of order $2^k$, dihedral group of order $2^k$, alternating group of degree $4$, symmetric group of degree $4$ or alternating group of degree $5$ effectively acting on the base of conic bundle. Also we construct infinitely many examples of such quotients which are…
For sufficiently ample linear systems on rational surfaces we show that a very general associated Brauer-Severi surface bundle is not stably rational.
Rationality of Brauer-Severi surface bundles over rational 3-folds
本文研究有理三维流形上Brauer-Severi曲面丛的稳定有理性问题，给出了其不是稳定有理的一个充分条件，并通过具体例子和构造一族一般成员光滑且非稳定有理的曲面丛加以验证。
We discuss the strong rational connectedness of smooth rationally connected surfaces. We prove in lots of cases, including the smooth locus of a log del Pezzo surface, the rational connectedness indeed implies the strong rational conectedness. This confirms a conjecture due to Hassett and Tschinkel in \cite{ht08}.
We prove the rationality of the coarse moduli spaces of Coble surfaces and of nodal Enriques surfaces over the field of complex numbers.
Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is non-singular, then this rational map is a morphism. As a consequence, we obtain that, if the surface is rational and we fix a divisor class containing a non-singular rational curve, then the scheme parametrizing rational curves on this class is irreducible. Further, if the class has non-negative self-intersection, then the scheme of rational curves has expecte…
In this paper, we show that the Bäcklund transformations of Painlevé equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlevé equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlevé equations.
We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves…
We provide a real analog of the Yau-Zaslow formula counting rational curves on $K3$ surfaces.
We proved the existence of rational curves in every linear system on a general K3 surface and that all rational curves in the hyperplane class are nodal on a general K3 surface of small genus.
We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log del Pezzo surface such that for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.
$G$-solid Rational Surfaces
本文在复数域上分类了 $G$-solid 有理曲面，给出了完整的分类结果。
We discuss the rational points on del Pezzo surface of degree 1 and 2 over any finite field $\mathbb F_q$, and give out the explicit equations of del Pezzo surfaces that have unique rational point.
We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a 9-dimensional vector space over an algebraically closed field of characteristic different from 2 and 3. To a general such element, one can naturally associate an Abelian surface X, which is embedded in 8-dimensional projective space. We study the combinatorial structure of this embedding and explicitly recover the genus 2 curve whose Jacobian variety is X. We also classify the types of degenerations of X that can occur. Taking the union over all Abelian surfaces in Heisenberg norma…
We prove that a very general projective K3 surface does not admit a dominant self rational map of degree at least two.
We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.
The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups : simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of certain pentagon in the Leech lattice and also in a complex algebraic geometry of $K3$ surfaces.
We answer a question of Ciro Ciliberto about cylinders in rational surfaces which are obtained by blowing up the plane at points in general position.
We prove that the Kawamata-Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W_2(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata-Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.
We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.
In this paper, we compare the moduli spaces of rank-3 vector bundles stable with respect to different ample divisors over rational ruled surfaces. We also discuss the irreducibility, unirationality, and rationality of these moduli spaces.
We study the distribution of algebraic points on K3 surfaces.
We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata-Viehweg vanishing.
We give examples of non-isotrivial K3 surfaces over complex function fields with Zariski-dense rational points and N'eron-Severi rank one.
We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on del Pezzo surfaces, this provides a categorical birational invariant for geometrically rational surfaces.
Approximating curves on real rational surfaces
本文给出实有理曲面上简单闭曲线可被光滑有理曲线逼近的拓扑充要条件，并研究了给定复自交数的逼近问题。
We study the dominant rational maps from a general surface in P^{3} to surfaces of general type. We prove restrictions on the target surfaces, and special properties of the rational maps. We show that for a small degree the general surface has no such map. Moreover a slight improvement of a result of Catanese, on the number of moduli of a surface of general type, is also obtained.
We prove that the Hilbert scheme of points on a normal quasi-projective surface with at worst rational double point singularities is irreducible.
We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.
We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root subsystems of E8.
We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic.
Given a curve C on a projective nonsingular rational surface S, over an algebraically closed field of characteristic zero, we are interested in the set Omega_C of linear systems Lambda on S satisfying C is in Lambda, dim Lambda > 0, and the general member of Lambda is a rational curve. The main result of the paper gives a complete description of Omega_C and, in particular, characterizes the curves C for which Omega_C is non empty.
We classify, up to some lattice-theoretic equivalence, all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.
We prove a general specialization theorem which implies stable irrationality for a wide class of quadric surface bundles over rational surfaces. As an application, we solve with the exception of two cases, the stable rationality problem for any very general complex projective quadric surface bundle over the projective plane, given by a symmetric matrix of homogeneous polynomials. Both exceptions degenerate over a plane sextic curve and the corresponding double cover is a K3 surface.
A smooth complex rational affine surface with uncountably many real forms
本文构造了一个光滑复有理仿射曲面，证明其具有不可数多个互不同构的实形式，揭示了实代数几何中实形式的丰富性与复杂性。
For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.
In this paper, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces.
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some fixed vector. Choosing an angle equal to $π/2$ we obtain a special instance of a~isophote -- the so called contour curve. While contours on rational ruled surfaces are rational curves, this is no longer true for the isophotes. Hence we will provide a formula for their genus. Moreover we will show that the only surfaces with a~rational generic contour are just …
A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely certain types of graphs. This allows to extend the classification of rational singularities to multiplicity 8. We also discuss the complexity of rational resolution graphs.
In this paper, combining the works of Miyanishi-Tsunoda and Keel-McKernan, we prove the log Castelnuovo's rationality criterion for smooth quasiprojective surfaces over complex numbers.
We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.
Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic model. We prove that they admit exactly one rational real algebraic model. This was known earlier only for the two-sphere, torus, projective plane and the Klein bottle.
We prove that the moduli spaces of K3 surfaces with non-symplectic involution are rational for four deformation types. With the previous results, this establishes the rationality of those moduli spaces except two classical cases.
We show that the number of generators of the n-th cotangent cohomology group (n >=2) is the same for all rational surface singularities Y. For a large class of rational surface singularities, including quotient singularities, this number is also the dimension. For them we obtain an explicit formula for the corresponding Poincare series.
We investigate configurations of rational double points with the total Milnor number 21 on supersingular $K3$ surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular $K3$ surfaces.
We discuss some relations of moduli of sheaves on rational surfaces by using universal extensions. These are a generalization of Maruyama's method to construct Uhlenbeck compactification of moduli of vector bundles.
The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. We use birational transformations to construct rational cuspidal curves with four cusps on the Hirzebruch surfaces and find associated results for these curves.
We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii) a general point is contained in no image of a Del Pezzo surface.
Real rational surfaces
本文综述实有理曲面的拓扑与双有理几何研究成果，涵盖其分类与结构性质。
We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems.
We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension $0$. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard number.
Let X be a smooth rational surface. We calculate a DG quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces.
We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.
We completely classify redundant blow-ups appearing in the theory of rational surfaces with big anticanonical divisor due to Sakai. In particular, we construct a rational surface with big anticanonical divisor which is not a minimal resolution of a del Pezzo surface with only rational singularities, which gives a negative answer to a question raised in a paper by Testa, Várilly-Alvarado, and Velasco.
It is proved that a smooth rational surface in projective four-space, which is ruled by cubics or quartics has degree at most 12. It is also proved that a smooth rational surface in projective four-space which is the image of Fn by a linear system with "simple base points" (see text) has degree at most 12.
We proved that the union of rational curves is dense on a very general K3 surface and the union of elliptic curves is dense in the 1st jet space of a very general K3 surface, both in the strong topology.
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