Cyclic coverings of a smooth curve and branch locus of the moduli space of smooth curves

1999 ◽  
pp. 183-196 ◽  
Author(s):  
Esteban Gómez González
Keyword(s):  
Moduli Space ◽  
Smooth Curve ◽  
Branch Locus ◽  
Smooth Curves ◽  
Cyclic Coverings

scholarly journals MODULI SPACES OF PARABOLIC HIGGS BUNDLES AND PARABOLIC K(D) PAIRS OVER SMOOTH CURVES: I

1996 ◽  
Vol 07 (05) ◽  
pp. 573-598 ◽  
Author(s):  
HANS U. BODEN ◽  
KÔJI YOKOGAWA
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Morse Theory ◽  
Smooth Curve ◽  
Higgs Bundles ◽  
Poincaré Polynomial ◽  
Smooth Curves ◽  
Connected Manifold ◽  
Parabolic Bundles ◽  
Simply Connected

This paper concerns the moduli spaces of rank-two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a non-compact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.

Quantitation of measurement error with Optimal Segments: basis for adaptive time course smoothing

1993 ◽  
Vol 264 (6) ◽  
pp. E902-E911 ◽  
Author(s):  
D. C. Bradley ◽  
G. M. Steil ◽  
R. N. Bergman
Keyword(s):  
Measurement Error ◽  
Measurement Errors ◽  
Time Course ◽  
Smooth Curve ◽  
Random Group ◽  
Smooth Curves ◽  
Original Curve ◽  
Wide Range ◽  
Data Points ◽  
Time Courses

We introduce a novel technique for estimating measurement error in time courses and other continuous curves. This error estimate is used to reconstruct the original (error-free) curve. The measurement error of the data is initially assumed, and the data are smoothed with "Optimal Segments" such that the smooth curve misses the data points by an average amount consistent with the assumed measurement error. Thus the differences between the smooth curve and the data points (the residuals) are tentatively assumed to represent the measurement error. This assumption is checked by testing the residuals for randomness. If the residuals are nonrandom, it is concluded that they do not resemble measurement error, and a new measurement error is assumed. This process continues reiteratively until a satisfactory (i.e., random) group of residuals is obtained. In this case the corresponding smooth curve is taken to represent the original curve. Monte Carlo simulations of selected typical situations demonstrated that this new method ("OOPSEG") estimates measurement error accurately and consistently in 30- and 15-point time courses (r = 0.91 and 0.78, respectively). Moreover, smooth curves calculated by OOPSEG were shown to accurately recreate (predict) original, error-free curves for a wide range of measurement errors (2-20%). We suggest that the ability to calculate measurement error and reconstruct the error-free shape of data curves has wide applicability in data analysis and experimental design.

scholarly journals HALF NIKULIN SURFACES AND MODULI OF PRYM CURVES

2019 ◽  
pp. 1-38
Author(s):  
Andreas Leopold Knutsen ◽  
Margherita Lelli-Chiesa ◽  
Alessandro Verra
Keyword(s):  
Irreducible Component ◽  
Moduli Space ◽  
Smooth Curve ◽  
Complete Picture ◽  
Boundary Points

Let ${\mathcal{F}}_{g}^{\mathbf{N}}$ be the moduli space of polarized Nikulin surfaces $(Y,H)$ of genus $g$ and let ${\mathcal{P}}_{g}^{\mathbf{N}}$ be the moduli of triples $(Y,H,C)$ , with $C\in |H|$ a smooth curve. We study the natural map $\unicode[STIX]{x1D712}_{g}:{\mathcal{P}}_{g}^{\mathbf{N}}\rightarrow {\mathcal{R}}_{g}$ , where ${\mathcal{R}}_{g}$ is the moduli space of Prym curves of genus $g$ . We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map $\unicode[STIX]{x1D712}_{g}$ and confirms some striking analogies between it and the Mukai map $m_{g}:{\mathcal{P}}_{g}\rightarrow {\mathcal{M}}_{g}$ for moduli of triples $(Y,H,C)$ , where $(Y,H)$ is any genus $g$ polarized $K3$ surface. The proof is by degeneration to boundary points of a partial compactification of ${\mathcal{F}}_{g}^{\mathbf{N}}$ . These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques.

scholarly journals Relative geodesics in bi-invariant Lie groups

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160619
Author(s):  
E. Zhang ◽  
L. Noakes
Keyword(s):  
Order Differential Equation ◽  
Homogeneous Spaces ◽  
Lagrange Equation ◽  
Smooth Curve ◽  
Curve Matching ◽  
Matching Problem ◽  
Smooth Curves ◽  
First Order ◽  
Finite Dimensional ◽  
Special Cases

Motivated by registration problems, this paper deals with a curve matching problem in homogeneous spaces. Let G be a connected finite-dimensional bi-invariant Lie group and K a closed subgroup. A smooth curve g in G is said to be admissible if it can transform two smooth curves f 1 and f 2 in G / K from one to the other. An ( f 1 , f 2 )- relative geodesic (Holm et al. 2013 Proc. R. Soc. A 469 , 20130297. ( doi:10.1098/rspa.2013.0297 )) is defined as a critical point of the total energy E ( g ) as g varies in the set of all ( f 1 , f 2 )-admissible curves. We obtain the Euler–Lagrange equation, a first-order differential equation, satisfied by a relative geodesic. Furthermore, the Euler–Lagrange equation is simplified for the case where G / K is globally symmetric. As a concrete example, relative geodesics are found for special cases where G is SO(3) and K is SO(2). As an application of discrepancy for curves in S 2 , we construct and study a new measure of non-congruency for constant speed curves in Euclidean 3-space. Numerical examples are given to illustrate results.

scholarly journals On the birational gonalities of smooth curves

2014 ◽  
Vol 68 (1) ◽  
Author(s):  
E. Ballico
Keyword(s):  
Positive Integer ◽  
Smooth Curve ◽  
Smooth Curves

AbstractLet C be a smooth curve of genus g. For each positive integer r the birational r-gonality s

scholarly journals Factorization of Generalized Theta Functions Revisited

2017 ◽  
Vol 24 (01) ◽  
pp. 1-52
Author(s):  
Xiaotao Sun
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Theta Functions ◽  
Vanishing Theorems ◽  
Smooth Curves ◽  
Canonical Line Bundle

This survey is based on my lectures given in the last few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorems, we show, without using vanishing theorems, a new observation that [Formula: see text] is independent of all of the choices for any smooth curves. The estimate of various codimensions and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

2017 ◽  
Vol 60 (1) ◽  
pp. 199-207
Author(s):  
RUBEN A. HIDALGO ◽  
SAÚL QUISPE
Keyword(s):  
Moduli Space ◽  
Singular Locus ◽  
Equivalence Classes ◽  
Rational Maps ◽  
Branch Locus ◽  
Cubic Curve ◽  
Dimension 2 ◽  
Holomorphic Automorphisms

AbstractMilnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.

scholarly journals ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF THE MODULI SPACE OF RIEMANN SURFACES OF GENUS 4

2010 ◽  
Vol 52 (2) ◽  
pp. 401-408 ◽  
Author(s):  
ANTONIO F. COSTA ◽  
MILAGROS IZQUIERDO
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Branch Locus

AbstractUsing uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

scholarly journals ON THE CONNECTEDNESS OF THE BRANCH LOCI OF NON-ORIENTABLE UNBORDERED KLEIN SURFACES OF LOW GENUS

2014 ◽  
Vol 57 (1) ◽  
pp. 211-230 ◽  
Author(s):  
E. BUJALANCE ◽  
J. J. ETAYO ◽  
E. MARTÍNEZ ◽  
B. SZEPIETOWSKI
Keyword(s):  
Moduli Space ◽  
Prime Order ◽  
Topological Conjugacy ◽  
Branch Locus ◽  
Klein Surfaces ◽  
Topological Genus

AbstractThis paper is devoted to determine the connectedness of the branch loci of the moduli space of non-orientable unbordered Klein surfaces. We obtain a result similar to Nielsen's in order to determine topological conjugacy of automorphisms of prime order on such surfaces. Using this result we prove that the branch locus is connected for surfaces of topological genus 4 and 5.

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