scholarly journals Correction to: Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials

2021 ◽  
Author(s):  
Dawei Chen ◽  
Martin Möller ◽  
Adrien Sauvaget ◽  
Don Zagier
Keyword(s):  
Moduli Spaces ◽  
Intersection Theory ◽  
Abelian Differentials

A Correction to this paper has been published: https://doi.org/10.1007/s00222-020-00969-4

scholarly journals Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials

2020 ◽  
Vol 222 (1) ◽  
pp. 283-373 ◽  
Author(s):  
Dawei Chen ◽  
Martin Möller ◽  
Adrien Sauvaget ◽  
Don Zagier
Keyword(s):  
Moduli Spaces ◽  
Intersection Theory ◽  
Abelian Differentials

scholarly journals GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES

2000 ◽  
Vol 11 (05) ◽  
pp. 637-663 ◽  
Author(s):  
TYLER J. JARVIS
Keyword(s):  
Line Bundle ◽  
Moduli Spaces ◽  
Disjoint Union ◽  
Algebraic Curves ◽  
Quantum Cohomology ◽  
Smooth Curve ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Universal Curve ◽  
Irreducible Components

This article treats various aspects of the geometry of the moduli [Formula: see text] of r-spin curves and its compactification [Formula: see text]. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdVr) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle [Formula: see text] on the universal curve over the stack of stable curves, there is a smooth stack [Formula: see text] of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism [Formula: see text]. In the special case that [Formula: see text] is the relative dualizing sheaf, then [Formula: see text] is the stack [Formula: see text] of r-spin curves. We construct a smooth compactification [Formula: see text] of the stack [Formula: see text], describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g>1, the compactified stack of spin curves [Formula: see text] and its coarse moduli space [Formula: see text] are irreducible, and when r is even and [Formula: see text] is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when [Formula: see text] is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [Formula: see text] [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

scholarly journals Tropical fans and the moduli spaces of tropical curves

2009 ◽  
Vol 145 (1) ◽  
pp. 173-195 ◽  
Author(s):  
Andreas Gathmann ◽  
Michael Kerber ◽  
Hannah Markwig
Keyword(s):  
Moduli Spaces ◽  
Building Blocks ◽  
Intersection Theory ◽  
Tropical Curves ◽  
Rigorous Definition ◽  
Definition Of ◽  
Tropical Varieties

AbstractWe give a rigorous definition of tropical fans (the ‘local building blocks for tropical varieties’) and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point; a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some ℝr) together with the evaluation and forgetful morphisms. Using our results this gives new, easy and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any ℝr) through given points are independent of the points.

scholarly journals On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

2019 ◽  
Vol 155 (12) ◽  
pp. 2354-2398
Author(s):  
Michael Magee
Keyword(s):  
Moduli Spaces ◽  
Natural Extension ◽  
Abelian Differentials

J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s $\frac{3}{16}$ theorem to moduli spaces of abelian differentials on surfaces of genus ${\geqslant}2$.

scholarly journals Tevelev degrees and Hurwitz moduli spaces

2021 ◽  
pp. 1-32
Author(s):  
A. CELA ◽  
R. PANDHARIPANDE ◽  
J. SCHMITT
Keyword(s):  
Scattering Amplitudes ◽  
Exact Solutions ◽  
Projective Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Intersection Theory ◽  
Dyck Paths ◽  
Simple Recursion ◽  
Almost All ◽  
Two Parameter

Abstract We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of $g^1_d$ ’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

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