Moduli Space Of Degree Research Articles

Distribution of postcritically finite polynomials III: Combinatorial continuity

In the first part of the present paper, we continue our study of distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of Misiurewicz parameters with prescribed combinatorics toward the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developped by Kiwi and Dujardin-Favre. In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the boundary of the Tricorn. We also establish an approximation property by Misiurewicz parameters in the spirit of the previous one. Finally, we answer a question of Kiwi, exhibiting in the moduli space of degree 4 polynomials, non-trivial Impression of specific combinatorics.

Mori's Program for the Moduli Space of Conics in Grassmannian

We complete Mori's program for Kontsevich's moduli space of degree $2$ stable maps to the Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial desingularization.

Categories of FI type: A unified approach to generalizing representation stability and character polynomials

Representation stability is a theory describing a way in which a sequence of representations of different groups is related, and essentially contains a finite amount of information. Starting with Church–Ellenberg–Farb's theory of FI-modules describing sequences of representations of the symmetric groups, we now have good theories for describing representations of other collections of groups such as finite general linear groups, classical Weyl groups, and Wreath products Sn≀G for a fixed finite group G. This paper attempts to uncover the mechanism that makes the various examples work, and offers an axiomatic approach that generates the essentials of such a theory: character polynomials and free modules that exhibit stabilization.We give sufficient conditions on a category C to admit such structure via the notion of categories of FI type. This class of categories includes the examples listed above, and extends further to new types of categories such as the categorical power FIm, whose modules encode sequences of representations of m-fold products of symmetric groups. The theory is applied in [9] to give homological and arithmetic stability theorems for various moduli spaces, e.g. the moduli space of degree n rational maps P1→Pm.

Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C 2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials. We deduce from that the equidistribution of hyperbolic parameters with (d − 1) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for C 1-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with (d − 1) distinct cycles with prescribed multiplier toward the bifurcation measure for any (d − 1) multipliers outside a pluripolar set.

Distribution of postcritically finite polynomials II: Speed of convergence

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C 2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials. We deduce from that the equidistribution of hyperbolic parameters with (d -- 1) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for C 1-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with (d -- 1) distinct cycles with prescribed multiplier toward the bifurcation measure for any (d -- 1) multipliers outside a pluripolar set.

Distribution of postcritically finite polynomials

We prove that Misiurewicz parameters with prescribed combinatorics and hyperbolic parameters with (d − 1) distinct attracting cycles with given multipliers are equidistributed with respect to the bifurcation measure in the moduli space of degree d complex polynomials. Our proof relies on Yuan’s equidistribution results of points of small heights, and uses in a crucial way Epstein’s transversality results.

Algebraic Independence of Multipliers of Periodic Orbits in the Space of Rational Maps of the Riemann Sphere

We consider the space of degree $n\ge 2$ rational maps of the Riemann sphere with $k$ distinct marked periodic orbits of given periods. First, we show that this space is irreducible. For $k=2n-2$ and with some mild restrictions on the periods of the marked periodic orbits, we show that the multipliers of these periodic orbits, considered as algebraic functions on the above mentioned space, are algebraically independent over $\mathbb C$. Equivalently, this means that at its generic point, the moduli space of degree $n$ rational maps can be locally parameterized by the multipliers of any $2n-2$ distinct periodic orbits, satisfying the above mentioned conditions on their periods. This work extends previous similar result (arXiv:1305.0867) obtained by the author for the case of complex polynomial maps.

Bicritical rational maps: Dynamical limits and ideal limit points of period m curves

Let denote the moduli space of degree d rational maps with exactly two critical points. In this paper we determine the ideal limit points of the curves consisting of all maps f having a period m cycle with multiplier η.

Lyapunov exponents, bifurcation currents and laminations in bifurcation loci

Bifurcation loci in the moduli space of degree d rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period n and multiplier 0 or e iθ. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree d = 2, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin exists.

A Presentation for the Chow Ring of

We give a presentation for the Chow ring of the moduli space of degree 2 stable maps from 2-pointed rational curves to the projective line. Also, integrals of all degree 4 monomials in the hyperplane pullbacks and boundary divisors of this ring are computed using equivariant intersection theory. Finally, the presentation is used to give a new computation of the (previously known) values of the genus zero, degree 2, 2-pointed gravitational correlators of the projective line.

A Googly Amplitude from the B-model in Twistor Space

Recently it has been proposed that gluon scattering amplitudes in gauge theory can be computed from the D-instanton expansion of the topological B-model on P^{3|4}, although only maximally helicity violating (MHV) amplitudes have so far been obtained from a direct B-model calculation. In this note we compute the simplest non-MHV gluon amplitudes (++--- and +-+--) from the B-model as an integral over the moduli space of degree 2 curves in P^{3|4} and find perfect agreement with Yang-Mills theory.

Enriques Surfaces, Analytic Discriminants, and Borcherds's Φ Function

In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form Φ on the moduli space of marked, polarized Enriques surface of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. In [JT 94] and [JT 96], we defined and studied a meromorphic (multi-valued) modular form of weight 2, which we call the K3 analytic discriminant, on the moduli space of marked, polarized, K3 surfaces of degree 2d; in certain cases, including when , where p k are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a determinant of the Laplacian on a polarized K3 surface with respect to the Calabi-Yau metric together with the L 2 norm of the image of the period map with respect to a properly scaled holomorphic two form. Since the universal cover of any Enriques surface is a K3 surface, we can restrict the K3 analytic discriminant to the moduli space of degree 2 Enriques surfaces. The main result of this paper is the observation that the square of our degree 2 analytic discriminant, viewed as a function on the moduli space of degree 2 Enriques surfaces, is equal to the Borcherd's Φ function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Moody algebras and elliptic curves, and suggests further connections with higher dimensional Calabi-Yau varieties, specifically those which can be realized as complete intersections in some, possibly weighted, projective space.

Moduli of Quaternionic Superminimal Immersions of 2-Spheres into Quaternionic Projective Spaces

The aim of this paper is to describe the moduli spaces of degree d quaternionic superminimal maps from 2-spheres to quaternionic projective spaces HPn. We show that such moduli spaces have the structure of projectivized fibre products and are connected quasi-projective varieties of dimension 2nd + 2n + 2. This generalizes known results for spaces of harmonic 2-spheres in S4.

Moduli spaces of degree d hypersurfaces in ℙn

Moduli spaces of degree d hypersurfaces in ℙn