We use stable maps, and their stable lifts to the Semple bundle variety of second-order curvilinear data, to calculate certain characteristic numbers for rational plane curves. These characteristic numbers involve first-order (tangency) and second-order (inflectional) conditions. Although they may be virtual, they may be used as inputs in an enumeratively significant formula for the number of rational curves having a triple contact with a specified plane curve and passing through 3d-3 general points.
This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P^2 or P^1 x P^1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.
We study the group of polynomial automorphisms of $\mathbb{C}^3$ (resp. birational self-maps of $\mathbb{P}^3_\mathbb{C}$) that preserve the contact structure.
We shortly describe the algorithms behind some of the functions provided by the Macaulay2 package MultiprojectiveVarieties, a package for multi-projective varieties and rational maps between them.
Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parametrizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such non trivial rational maps, that is of topological degree >1.
Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results.
Let X be a compact nonsingular real algebraic variety. We prove that if a continuous map from X into the unit p-sphere is homotopic to a continuous rational map, then, under certain assumptions, it can be approximated in the compact-open topology by continuous rational maps. As a byproduct, we also obtain some results on approximation of smooth submanifolds by nonsingular subvarieties.
In this paper we show that if two central simple $k$-algebras generate the same cyclic subgroup in $\mathrm{Br}(k)$, then there are rational maps between varieties associated to these algebras, such as Brauer--Severi varieties, norm hypersurfaces and symmetric powers. In some cases we even have rational embeddings. We also relate the obtained results to the Amitsur conjecture.
In this paper we investigate the existence of generically finite dominant rational maps from products of curves to surfaces of general type. We prove that the product CxD of two distinct very general curves of genus g>6 and g'>1 does not admit dominant rational maps on other surfaces of general type.
The aim of this note is to give simple proofs of some results of Reichstein and Youssin (math.AG/9903162) about the behaviour of fixed points of finite group actions under rational maps. Our proofs work in any characteristic. We also give a short proof of the Nishimura lemma.
Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim of this paper is to generalized this result to irreducible rational homogeneous space of higher Picard number.
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.
Complex projective algebraic varieties with $\mathbb{C}^*$-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a $\mathbb{C}^*$-action with no proper isotropy subgroups.
The Macaulay2 package Cremona performs some computations on rational and birational maps between irreducible projective varieties. For instance, it provides methods to compute degrees and projective degrees of rational maps without any theoretical limitation, from which is derived a general method to compute the push-forward to projective space of Segre classes. Moreover, the computations can be done both deterministically and probabilistically. We give here a brief description of the methods and algorithms implemented.
This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus $g\geq 7$ and $g'\geq 3$ does not admit dominant rational maps of degree $> 1$ if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve.
We give a bound for the number of rational maps between algebraic varieties of general type under mild hypothesis on the canonical map. We use an idea inspired by Tanabe's work. Instead of attaching a morphism of Hodge structures to a rational map we simply associate to it a piece of the integral Hodge lattice. This procedure does not give an injective map, but by means of a geometric argument, we can estimate the number of maps with the same image.
With any \emph{surjective rational map} $f: \mathbb{P}^n \dashrightarrow \mathbb{P}^n$ of the projective space we associate a numerical invariant (\emph{ML degree}) and compute it in terms of a naturally defined vector bundle $E_f \longrightarrow \mathbb{P}^n$.
The purpose of this short note is to study dominant rational maps from punctual Hilbert schemes of length $k>1$ of projective K3 surfaces $S$ containing infinitely many rational curves. Precisely, we prove that their image is necessarily rationally connected if this rational map is not generically finite. As an application, we simplify the proof of C. Voisin's of the fact that symplectic involutions of any projective K3 surface $S$ act trivially on $\mathrm{CH}_0(S)$.
We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve, while the second is obtained as the intersection of the Abel section of the universal Jacobian with the zero section. Our comparison extends results of Cavalieri-Marcus-Wise where the same identity was proved over on the locus of rational tails curves.
The dominant rational maps of finite degree from a fixed variety to varieties of general type, up to birational isomorphisms, form a finite set. This has been known as the Iitaka-Severi conjecture, and is nowdays an established result, in virtue of some recent advances in the theory of pluricanonical maps. We study the question of finding some effective estimate for the finite number of maps, and to this aim we provide some update and refinement of the classical treatment of the subject.
In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing the natural parametrization of maps by means of the space of linear projections in a suitable projective space, and this leads to some new insight in the geometry of the finiteness theorem.
We prove a dynamical Shafarevich theorem on the finiteness of the set of isomorphism classes of rational maps with fixed degeneracies. More precisely, fix an integer d at least 2 and let K be either a number field or the function field of a curve X over a field k, where k is of characteristic zero or p>2d-2 that is either algebraically closed or finite. Let S be a finite set of places of K. We prove the finiteness of the set of isomorphism classes of rational maps over K with a natural kind of good reduction outside of S. We also prove auxiliary results on finiteness of reduced effective divis…
We study dominant rational maps from a product of two curves to surfaces with $p_{g} = q = 0$. Given two curves which satisfy a mild genericity assumption and have large genus relative to their gonality, we show that the degree of irrationality of their product is equal to the product of their gonalities. Moreover, we prove that the degree of irrationality of a product of two hyperelliptic curves is 4.
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the firs…
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of the canonical contact structure, for each member of this family.
A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus {\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small…
We give a brief survey on the entropy of holomorphic self maps $f$ of compact Kähler manifolds, and rational dominating self maps $f$ of smooth projective varieties. We emphasize the connection between the entropy and the spectral radii of the induced action of $f$ on the homology of the compact manifold. The main conjecture for the rational maps states that modulo birational isomorphism all various notions of entropy and the spectral radii are equal.
We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.
Let $\Hol_{x_0}^{\bf n} (\C¶^1, X)$ be the space of based holomorphic maps of degree ${\bf n}$ from $\C¶^1$ into a simply connected algebraic variety $X$. Under some condition we prove that the map $\map \Hol_{x_0}^{\bf n} (\C¶^1, X). \Hol_{x_0}^{d{\bf n}} (\C¶^1, X).$ obtained by compositing $f \in \Hol_{x_0}^{\bf n} (\C¶^1, X)$ with $g(z)=z^d, z \in \C¶^1$ induces rational homotopy equivalence up to some dimension, which tends to infinity as the degree grows.
We show that an $n$-dimensional compact Kähler manifold $X$ admitting a non-degenerate meromorphic map $f:{\bf C}^n\to X$ of order $ρ_f<2$ is rationally connected.
For a finite dimensional vector space equipped with a $\mathbb C$-algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps.
Let F : W --> V be a dominant rational map between quasi-projective varieties of the same dimension. We give two proofs that h_V(F(P)) >> h_W(P) for all points P in a nonempty Zariski open subset of W. For dominant rational maps F : P^n --> P^n, we give a uniform estimate in which the implied constant depends only on n and the degree of F. As an application, we prove a specialization theorem for equidimensional dominant rational maps to semiabelian varieties, providing a complement to Habegger's recent theorem on unlikely intersections.
One considers the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian integral domain.
We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable sense.
Given $r>n$ general hyperplanes in $\mathbb P^n,$ a star configuration of points is the set of all the $n$-wise intersection of them. We introduce {\it contact star configurations}, which are star configurations where all the hyperplanes are osculating to the same rational normal curve. In this paper we find a relation between this construction and Hadamard products of linear varieties. Moreover, we study the union of contact star configurations on a same conic in $\mathbb P^2$, we prove that the union of two contact star configurations has a special $h$-vector and, in some cases, this is a co…
We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational classification of singular holomorphic foliations on surfaces, and the dynamics of holomorphic maps. Following this analogy, we introduce the notion of minimal holomorphic model for holomorphic maps. We give sufficient conditions which ensure the uniqueness of such a model.
We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the "saturated special fiber ring", which turns out to be a fundamental tool to analyze the degree of a rational map. We …
We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\mathscr F_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space $\mathbb P^r$, when $1\le q \le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.
We define an algebro-geometric model for the space of rational maps from a smooth curve X to an algebraic group G, and show that this space is homologically contractible. As a consequence, we deduce that the moduli space Bun(G) of G-bundles on X is uniformized by the appropriate rational version of the affine Grassmannian, where the uniformizing map has contractible fibers.
We use intuitive results from algebraic topology and intersection theory to clarify the pullback action on cohomology by compositions of rational maps. We use these techniques to prove a simple sufficient criterion for functoriality of a composition of two rational maps on all degrees of cohomology and we then reprove the criteria of Diller-Favre, Bedford-Kim, and Dinh-Sibony. We conclude with a cautionary example.
Pop proved that a smooth curve C over an ample field K that has a K-rational point has |K| many K-rational points. We strengthen this result by showing that there are |K| many K-rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfect ample field has no existentially definable proper infinite subfields.
We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact Riemann surface of genus at least two have rational singularities. This has consequences for the representation zeta function of the special linear group over the integers and over the p-adic integers.
Consider a rational map from a projective space to a product of projective spaces, induced by a collection of linear projections. Motivated by the the theory of limit linear series and Abel-Jacobi maps, we study the basic properties of the closure of the image of the rational map using a combination of techniques of moduli functors and initial degenerations. We first give a formula of multi-degree in terms of the dimensions of intersections of linear subspaces and then prove that it is Cohen-Macaulay. Finally, we compute its Hilbert polynomials.
We construct the moduli space, $M_d$, of degree $d$ rational maps on $\mathbb{P}^1$ in terms of invariants of binary forms. We apply this construction to give explicit invariants and equations for $M_3$. Using classical invariant theory, we give solutions to the following problems: (1) explicitly construct, from a moduli point $P\in M_d(k)$, a rational map $ϕ$ with the given moduli; (2) find a model for $ϕ$ over the field of definition (i.e. explicit descent). We work out the method in detail for the cases $d=2,3$.
