Maps Of Genus Research Articles

Divisors and curves on logarithmic mapping spaces

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime

We classify compact Riemann surfaces of genus \(g\), where \(g-1\) is a prime \(p\), which have a group of automorphisms of order \(\rho(g-1)\) for some integer \(\rho\ge 1\), and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for \(\rho>6\), and of the first and third authors for \(\rho=\) 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus \(p+1\), together with the non-orientable regular hypermaps of characteristic \(-p\), with automorphism group of order divisible by the prime \(p\); this extends results of Conder, Siraň and Tucker for maps.

Moduli of stable maps in genus one and logarithmic geometry, I

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to the log minimal model program for $\mathcal{M}_{1,n}$. We construct explicit factorizations of the rational maps among Smyth's modular compactifications of pointed elliptic curves.

Moduli of stable maps in genus one and logarithmic geometry, II

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give a new infinite family of such maps in genus 3.

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.

Tautological rings of stable map spaces

We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is P n , these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 11.

Topological strings in generalized complex space

A two-dimensional topological sigma-model on a generalized Calabi-Yau target space $X$ is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure $J$ and a pure spinor $\rho$ on $X$. In the present construction the algebra of $Q$-transformations automatically closes off-shell, the model transparently depends only on $J$, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N=2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector $\beta$ and recover holomorphic noncommutative Kontsevich $*$-product.

Map Genus, Forbidden Maps, and Monadic Second-Order Logic

A map is a graph equipped with a circular order of edges around each vertex. These circular orders represent local planar embeddings. The genus of a map is the minimal genus of an orientable surface in which it can be embedded. The maps of genus at most $g$ are characterized by finitely many forbidden maps, relatively to an appropriate ordering related to the minor ordering of graphs. This yields a "noninformative" characterization of these maps, that is expressible in monadic second-order logic. We give another one, which is more informative in the sense that it specifies the relevant surface embedding, in addition to stating its existence.

Associativity Relations in Quantum Cohomology

The geometry of moduli spaces of stable maps of genus 0 curves into a complex projective manifold X leads to a system of quadratic equations in the tree-level (genus 0) Gromov-Witten numbers of X. In elementary examples, these equations solve for all such numbers, uniquely and consistently, from starting data. The beautiful paper of Di Francesco and Itzykson [1] presents a number of examples in this context. One of the foundational papers in the area of quantum cohomology, [4], explains this phenomenon, at least in some cases, by proving the first reconstruction theorem. This theorem applies to manifoldsX such thatH∗(X,Q) is generated byH(X). This result gives an effective procedure for solving for genus 0 Gromov-Witten numbers from starting data using the quadratic relations (since this entire paper is concerned only with the genus 0 invariants, we omit explicit mention of genus from now on). In all but the simplest cases there will be more than one way of using the relations to solve for the numbers. In other words, the system of equations is overdetermined. In the same paper the authors ask whether the seemingly redundant equations follow algebraically from the useful ones. Consistency of this overdetermined system of equations, as an algebraic (or combinatorial) statement rather than a geometric statement, has only been noted in the literature in isolated instances, cf. [2]. The equations were predicted on the basis of physical theories and later confirmed by rigorous study of the moduli spaces. The view taken by the physicists is quite useful: the numbers are combined into a generating function and the relations are represented by differential equations. The survey paper [6] documents these early studies. This paper continues in the spirit of these early investigations into the structure of the relations. The main result is a generalization of the first reconstruction theorem. Keeping the hypothesis that the cohomology ring of X is generated by divisors, we show that an initial collection of numbers and relations determines, purely algebraically, the entire system of relations (strong reconstruction) as well as the Gromov-Witten numbers. Examples, in the last section, illustrate the existence of non-geometric solutions. For a manifold X with H = H∗(X,C), the associativity relations can be expressed geometrically by saying that an associated connection on TH is flat, i.e., that the quantum potential function dictates on H the structure of a Frobenius manifold [2, 4]. Near a semisimple solution, the equations for flatness can be proved equivalent (with suitable assumptions) to the well-studied Schlesinger equations, cf. [5]. Strong

Kepler-Poinsot-Type Realizations of Regular Maps of Klein, Fricke, Gordan and Sherk

AbstractThe paper describes polyhedral realizations for Felix Klein's map {3, 7}8 of genus 3, for Gordan's map {4, 5}6 of genus 4, and for two maps of genus 5, the Klein-Fricke map of type {3, 8} and Sherk's map of type {4, 6}. The polyhedra have self-intersections but high symmetry and thus are close analogues to the Kepler-Poinsot-polyhedra.

On weakly neighborly polyhedral maps of arbitrary genus

A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We lay the foundation for an investigation of the w.n.p. maps of arbitrary genus. In particular we find all the w.n.p. maps of genus 0, we prove that for everyg> the number of w.n.p. maps of genusg (orientable or not) is finite, and we find an upper bound for the number of vertices in a w.n.p. map of genusg>0. This upper bound grows as (4g(2/3) wheng→∞.