scholarly journals A MODULI STACK OF TROPICAL CURVES

2020 ◽  
Vol 8 ◽  
Author(s):  
RENZO CAVALIERI ◽  
MELODY CHAN ◽  
MARTIN ULIRSCH ◽  
JONATHAN WISE
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curves ◽  
Tropical Geometry ◽  
Moduli Space Of Curves ◽  
Tropical Curves ◽  
Moduli Spaces Of Curves ◽  
Moduli Stack ◽  
Marked Points ◽  
Moduli Problems

We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

scholarly journals MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY

2016 ◽  
Vol 4 ◽  
Author(s):  
RENZO CAVALIERI ◽  
SIMON HAMPE ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Tropical Geometry ◽  
Stable Curves ◽  
Tropical Curves ◽  
Graphic Matroid ◽  
Fibre Products ◽  
Algebraic Counterpart

We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

scholarly journals A computational approach to the ample cone of moduli spaces of curves

2018 ◽  
Vol 28 (01) ◽  
pp. 37-51
Author(s):  
Claudio Fontanari ◽  
Riccardo Ghiloni ◽  
Paolo Lella
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Rational Curves ◽  
Computational Approach ◽  
Alternate Proof ◽  
Moduli Spaces Of Curves ◽  
Ample Cone ◽  
Marked Points

We present an alternate proof, much quicker and more straightforward than the original one, of the celebrated F-conjecture on the ample cone of the moduli space [Formula: see text] of stable rational curves with [Formula: see text] marked points in the case [Formula: see text].

scholarly journals Zero cycles on the moduli space of curves

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Rahul Pandharipande ◽  
Johannes Schmitt
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Rational Surface ◽  
K3 Surfaces ◽  
Moduli Of Curves ◽  
K3 Surface ◽  
Moduli Space Of Curves ◽  
Moduli Spaces Of Curves ◽  
Open Questions ◽  
Witten Theory

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed. Comment: Published version

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 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.

scholarly journals Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

2014 ◽  
Vol 17 (A) ◽  
pp. 128-147 ◽  
Author(s):  
Reynald Lercier ◽  
Christophe Ritzenthaler ◽  
Florent Rovetta ◽  
Jeroen Sijsling
Keyword(s):  
Finite Fields ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Moduli Space Of Curves ◽  
Isomorphism Classes ◽  
Families Of Curves ◽  
Plane Quartics ◽  
Smooth Plane

AbstractWe study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

TWO REMARKS ON SUBVARIETIES OF MODULI SPACES

2008 ◽  
Vol 19 (02) ◽  
pp. 237-243 ◽  
Author(s):  
KIRTI JOSHI
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
Moduli Space Of Curves ◽  
Rank Zero ◽  
Zero Locus

We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.

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