scholarly journals Components with the expected codimension in the moduli scheme of stable spin curves

2015 ◽  
Vol 69 (1) ◽  
pp. 1
Author(s):  
Edoardo Ballico
Keyword(s):  
Line Bundles ◽  
Stable Model ◽  
Stable Curves ◽  
Theta Characteristics

Here we study the Brill–Noether theory of “extremal” Cornalba’s theta-characteristics on stable curves C of genus g, where “extremal” means that they are line bundles on a quasi-stable model of C with #(Sing(C)) exceptional components.

scholarly journals Components with the expected codimension in the moduli scheme of stable spin curves

2015 ◽  
Vol 69 (1) ◽  
pp. 1-4
Author(s):  
Edoardo Ballico
Keyword(s):  
Line Bundles ◽  
Stable Model ◽  
Stable Curves ◽  
Theta Characteristics

AbstractHere we study the Brill-Noether theory of “extremal” Cornalba’s theta-characteristics on stable curves C of genus g, where “extremal” means that they are line bundles on a quasi-stable model of C with #(Sing(C)) exceptional components

scholarly journals Towards an enumerative geometry of the moduli space of twisted curves and rth roots

2008 ◽  
Vol 144 (6) ◽  
pp. 1461-1496 ◽  
Author(s):  
Alessandro Chiodo
Keyword(s):  
Moduli Space ◽  
Chern Character ◽  
Direct Image ◽  
Enumerative Geometry ◽  
Line Bundles ◽  
Stable Curves ◽  
Moduli Stack ◽  
Definition Of ◽  
Standard Techniques

AbstractThe enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.

scholarly journals A note on the theta characteristics of a compact Riemann surface

2004 ◽  
Vol 76 (3) ◽  
pp. 415-424
Author(s):  
Indranil Biswas
Keyword(s):  
Riemann Surface ◽  
Quadratic Form ◽  
Line Bundle ◽  
Vector Fields ◽  
Compact Riemann Surface ◽  
Line Bundles ◽  
Square Root ◽  
Topological Interpretation ◽  
Theta Characteristics

AbstractLet X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.

scholarly journals Gromov compactness in non-archimedean analytic geometry

2018 ◽  
Vol 2018 (741) ◽  
pp. 179-210 ◽  
Author(s):  
Tony Yue Yu
Keyword(s):  
Symplectic Geometry ◽  
Compactness Theorem ◽  
Holomorphic Curves ◽  
Line Bundles ◽  
Analytic Geometry ◽  
Kähler Structure ◽  
Stable Curves ◽  
Berkovich Spaces ◽  
Moduli Stack ◽  
Bounded Area

Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

A Study of the Planning of Water Pollution Control for Second Songhua River in China

1991 ◽  
Vol 23 (1-3) ◽  
pp. 41-48
Author(s):  
Yin Jun
Keyword(s):  
Water Pollution ◽  
Mathematical Model ◽  
Pollution Control ◽  
Sewage Treatment Plant ◽  
Treatment Plant ◽  
Initial Structure ◽  
Stable Model ◽  
Control Plan ◽  
Songhua River ◽  
The Second Songhua River

The paper takes the Second Songhua River as an object for research and selects Thomas's BOD-DO stable model as the initial structure on the basis of overall investigations and analyses on water pollution in every reach. In view of the characteristics of the river being located at the north, values k’1, k’2 and k’3 in dry season of winter were determined and calculated, and a series analyses have been made. The self-purification ability of the river and the total elimination amount of the main pollutants BOD5 were also calculated. In order to minimize the required cost, we distributed the cost to the main pollution sources, which are to be controlled. We firstly set a cost function of sewage treatment plant by series design and calculated the related cost parameters, then calculated two kinds of optimal distributing models of BOD5 elimination, which were a mathematical model of extreme value of conditions and a matrix mathematical model. Now they have been applied to the practical pollution control plan for the Second Songhua River.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

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