An explicit formula of the normalized Mumford form

Suppose X = X g is a compact orientable surface of genus g ≥ 2 . A cyclic group Z p of prime order p of orientation-preserving homeomorphisms of X is called finitely maximal if there is no proper finite supergroup G ≤ Homeo + ( X ) containing Z p . In [Maximal group actions on compact oriented surfaces, J. Algebra 472 (2017) 1–14], V. Peterson, J. Russell, and A. Wootton showed that if Z p is finitely maximal then the number r of fixed points of Z p is maximal, or equivalently the genus h of X / Z p is minimal. Moreover, they exhibited an infinite sequence of genera within which any given action of Z p on X implies Z p is contained in some finite supergroup and demonstrate for genera outside of this sequence the existence of at least one finitely maximal Z p -action. Here we show that the set of equivalence classes of finitely maximal cyclic actions of a fixed Teichmüller dimension d defined as 3 ( h − 1 ) + r is combinatorially dense, in a suitable sense, in the set of classes of all actions of prime order of such dimension. This is a bit of a surprising result in the context of the results of Peterson, Russell and Wootton and it has some consequences concerning the singular locus of the moduli space of algebraic curves of a given genus, which we briefly describe at the end of the Introduction.

In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics. This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for $\omega$ -admissible metrized line bundles depend on $\omega$ . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric on the moduli space is algebraic.

Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image $\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex $\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ... \tens L^{[n]})$ in terms of of the hyperderived spectral sequence $E^{p,q}_1$ associated to the derived $k$-fold tensor power of the complex $\comp{\mc{C}}_L$. The study of the $\perm_n$-invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of $X^{[n]}$ with values in $L^{[n]} \tens L^{[n]}$ and $\Lambda^k L^{[n]}$.

This paper develops a framework for shape analysis of tree-like structures with the following common features: (1) a main branch viewed as a parameterized curve in R3, and (2) a random number of secondary branches, each one of them a parameterized curve in R3, emanating from the main branch at arbitrary points. In this framework, comparisons of objects is based on shapes-scales-orientations of the curves involved, and locations and number of the side branches. The objects are represented as compos- ite curves made up of: a main branch and a continuum of side branches along the main branch with each branch being a curve in R3 itself (including the null curve, or zero curve). Extending the previous work on elastic shape analysis of Euclidean curves, the space of these composite curves is endowed with a natural Riemannian metric, using the SRVF representation, and one computes geodesic paths in the quotient space of this representation modulo the re-parameterization function. As a result, appropriate geomet- ric structures are optimally matched across trees, and geodesic paths show deformations of main branches into each other while either deforming/sliding/creating/destroying the side branches. We present some preliminary results using axonal trees taken from the Neuromorpho database.

Normal generation of line bundles on multiple coverings

In this paper we construct Ulrich bundles of low rank on three-dimensional scrolls (with respect to the tautological line bundle). We pay special attention to the four types of three-fold scrolls in ${\mathbb{P}}^5$, which were classified in [31].

This chapter uses techniques from the theory of local dynamics of holomorphic germs tangent to the identity to prove three index theorems for global meromorphic maps of projective space. More precisely, the chapter seeks to prove a particular index theorem: Let f : ℙⁿ ⇢ ℙⁿ be a meromorphic self-map of degree ν‎ + 1 ≥ 2 of the complex n-dimensional projective space. Let Σ‎(f) = Fix(f) ∪ I(f) be the union of the indeterminacy set I(f) of f and the fixed points set Fix(f) of f. Let Σ‎(f) = ⊔subscript Greek Small Letter AlphaΣ‎subscript Greek Small Letter Alpha be the decomposition of Σ‎ in connected components, and denote by N the tautological line bundle of ℙⁿ. After laying out the statements under this theorem, the chapter discusses the proofs.

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

The super Mumford form and Sato Grassmannian

This paper discusses a solution to the problem of how to approximate an algebraic curve of any degree with piecewise parametric curves. The approximation can be performed using any differentiable parametric curve form. An implementation using rational cubic Bezier curves results in a G 2 piecewise approximation which experiences better than O(h 7) convergence. If the algebraic curve happens to be a cubic curve of genus zero, a parametric curve can be found which exactly fits the given algebraic curve to within floating point precision. An error bound is developed which is much tighter than previously published bounds.

The expression [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] transforms as a symmetric (0, 2) tensor under projective coordinate changes of a domain in R n so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold, the section u can be regarded as a metric potential analogous to the local potential in Kählergeometry. Let M be a compact locally projectively flat manifold. We prove that if u is a negative section of the dual of the tautological bundle such that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] is a Riemannian metric, then M is projectively equivalent to a quotient of a bounded convex domain in R n . The same is true for such manifolds M with boundary if u = 0 on the boundary. This theorem is an analog of a result of Schoen and Yau in locally conformally flat geometry. The proof involves affine differential geometry techniques developed by Cheng and Yau.

This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schrödinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection between integrability and the self-dual Yang-Mills equations, and describes the correspondence between solutions to integrable equations and holomorphic vector bundles over twistor space.

For smooth families $\\mathcal{X} \\to S$ of projective algebraic curves and holomorphic line bundles $\\mathcal{L, M} \\to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\\langle \\mathcal{L, M} \\rangle \\to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\\mathcal{L}$ and $\\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.

Equivalence between the following categories is established: 1) A category of arbitrary vector bundles on algebraic curves defined over a field of arbitrary characteristic, and 2) a category of infinite dimensional vector spaces corresponding to certain points of Grassmannians together with their stabilizers. Our contravariant functor between these categories gives a full generalization of the well-known Krichever map, which assigns points of Grassmannians to the geometric data consisting of curves and line bundles. As an application, a solution to the classical problem of Wallenberg-Schur of classifying all commutative algebras consisting of ordinary differential operators is obtained. It is also shown that the KP flows produce all generic vector bundles on arbitrary algebraic curves of genus greater than one.

We determine the effective cone of the Quot scheme parametrizing all rank r, degree d quotient sheaves of the trivial bundle of rank n on $${\mathbb{P}^1}$$ . More specifically, we explicitly construct two effective divisors which span the effective cone, and we also express their classes in the Picard group in terms of a known basis.