Etale homotopy type of the moduli spaces of algebraic curves

1997 ◽  
pp. 85-96 ◽  
Author(s):  
Takayuki Oda
Keyword(s):  
Homotopy Type ◽  
Moduli Spaces ◽  
Algebraic Curves

scholarly journals Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

2009 ◽  
Vol 37 (8) ◽  
pp. 2649-2678 ◽  
Author(s):  
S. B. Bradlow ◽  
O. García-Prada ◽  
V. Mercat ◽  
V. Muñoz ◽  
P. E. Newstead
Keyword(s):  
Moduli Spaces ◽  
Algebraic Curves ◽  
Coherent Systems

scholarly journals ON THE GEOMETRY OF MODULI SPACES OF COHERENT SYSTEMS ON ALGEBRAIC CURVES

2007 ◽  
Vol 18 (04) ◽  
pp. 411-453 ◽  
Author(s):  
S. B. BRADLOW ◽  
O. GARCÍA-PRADA ◽  
V. MERCAT ◽  
V. MUÑOZ ◽  
P. E. NEWSTEAD
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curves ◽  
Homotopy Groups ◽  
Coherent System ◽  
Coherent Systems ◽  
Picard Groups ◽  
Algebraic Vector ◽  
The Stability ◽  
Almost All

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for different values of α when k ≤ n and the variation of the moduli spaces when we vary α. As a consequence, for sufficiently large α, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n - 1 explicitly, and give the Poincaré polynomials for the case k = n - 2. In an appendix, we describe the geometry of the "flips" which take place at critical values of α in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD (n,d,k) = 1.

scholarly journals Tau Function and Moduli of Meromorphic Quadratic Differential

2022 ◽  
Author(s):  
Dmitry Korotkin ◽  
◽  
Peter Zograf ◽  
Keyword(s):  
Moduli Spaces ◽  
Quadratic Differential ◽  
Algebraic Curves ◽  
Picard Group ◽  
Quadratic Differentials ◽  
Tau Function ◽  
Tau Functions

The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

scholarly journals Correspondence theorems via tropicalizations of moduli spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550043 ◽  
Author(s):  
Andreas Gross
Keyword(s):  
Moduli Spaces ◽  
Toric Variety ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Tropical Curves ◽  
Algebraic Tori ◽  
Pass Through ◽  
General Correspondence ◽  
Correspondence Theorem ◽  
Generic Points

We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind, we prove a general correspondence theorem for enumerative problems which are defined via “evaluation maps” in both the algebraic and tropical world. Applying this to our motivational example, we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.

Moduli spaces of algebraic curves with rational maps

1975 ◽  
Vol 78 (2) ◽  
pp. 283-292 ◽  
Author(s):  
Herbert Lange
Keyword(s):  
Algebraic Variety ◽  
Moduli Spaces ◽  
Algebraic Curves ◽  
Rational Maps ◽  
Closed Field ◽  
Rational Map ◽  
Algebraically Closed Field

Let ℳg be the coarse moduli scheme of curves of genus g. For an algebraically closed field k define is a quasiprojective algebraic variety over k, its dimension being 3g – 3 for g ≥ 2, 1 for g = 1, and 0 for g = 0. It can be considered as the moduli variety for the classes of birationally equivalent curves of genus g over k. For 0 < g, g′ and n ≥ 1 let be the subset of those points of whose corresponding curves possess a rational map of degree n into a curve of genus g′ over k.

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].

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