scholarly journals On the unirationality of moduli spaces of pointed curves

2021 ◽  
Author(s):  
Hanieh Keneshlou ◽  
Fabio Tanturri
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Smooth Curves ◽  
Projective Curves ◽  
G 12 ◽  
G 11 ◽  
Marked Points ◽  
G 13

AbstractWe show that$$\mathcal {M}_{g,n}$$Mg,n, the moduli space of smooth curves of genusgtogether withnmarked points, is unirational for$$g=12$$g=12and$$2 \le n\le 4$$2≤n≤4and for$$g=13$$g=13and$$1 \le n \le 3$$1≤n≤3, by constructing suitable dominant families of projective curves in$$\mathbb {P}^1 \times \mathbb {P}^2$$P1×P2and$$\mathbb {P}^3$$P3respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genusgtogether withnunordered points, establishing their unirationality for$$g=11, n=7$$g=11,n=7and$$g=12, n =5,6$$g=12,n=5,6.

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.

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].

Chow rings of Mp̃0,2(N,d) and M¯0,2(ℙN−1,d) and Gromov–Witten invariants of projective hypersurfaces of degree 1 and 2

2017 ◽  
Vol 28 (12) ◽  
pp. 1750090 ◽  
Author(s):  
Hayato Saito
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Intersection Number ◽  
Witten Invariant ◽  
Chow Ring ◽  
Stable Curves ◽  
Chow Rings ◽  
Number Formula ◽  
Genus Formula ◽  
Marked Points

In this paper, we prove formulas that represent two-pointed Gromov–Witten invariant [Formula: see text] of projective hypersurfaces with [Formula: see text] in terms of Chow ring of [Formula: see text], the moduli spaces of stable maps from genus [Formula: see text] stable curves to projective space [Formula: see text]. Our formulas are based on representation of the intersection number [Formula: see text], which was introduced by Jinzenji, in terms of Chow ring of [Formula: see text], the moduli space of quasi maps from [Formula: see text] to [Formula: see text] with two marked points. In order to prove our formulas, we use the results on Chow ring of [Formula: see text], that were derived by Mustaţǎ and Mustaţǎ. We also present explicit toric data of [Formula: see text] and prove relations of Chow ring of [Formula: see text].

TORELLI THEOREM FOR MODULI SPACES OF SL(r,ℂ)-CONNECTIONS ON A COMPACT RIEMANN SURFACE

2009 ◽  
Vol 11 (01) ◽  
pp. 1-26
Author(s):  
INDRANIL BISWAS ◽  
VICENTE MUÑOZ
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Complex Structure ◽  
Real Numbers ◽  
Stable Vector Bundles ◽  
Complex Variety ◽  
Projective Curves ◽  
Torelli Theorem

Let X be any compact connected Riemann surface of genus g, with g ≥ 3. For any r ≥ 2, let [Formula: see text] denote the moduli space of holomorphic SL (r,ℂ)-connections over X. It is known that the biholomorphism class of the complex variety [Formula: see text] is independent of the complex structure of X. If g = 3, then we assume that r ≥ 3. We prove that the isomorphism class of the variety [Formula: see text] determines the Riemann surface X uniquely up to an isomorphism. A similar result is proved for the moduli space of holomorphic GL (r,ℂ)-connections on X. We also show that the Torelli theorem remains valid for the moduli spaces of connections, as well as those of stable vector bundles, on geometrically irreducible smooth projective curves defined over the field of real numbers.

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.

Hitchin pairs on reducible curves

2018 ◽  
Vol 29 (03) ◽  
pp. 1850015 ◽  
Author(s):  
Usha N. Bhosle
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Disjoint Union ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Birational Morphism ◽  
Smooth Curves ◽  
Irreducible Components ◽  
Proper Map ◽  
The Relationship

We define semistable generalized parabolic Hitchin pairs (GPH) on a disjoint union [Formula: see text] of integral smooth curves and construct their moduli spaces. We define a Hitchin map on the moduli space of GPH and show that it is a proper map. We construct moduli spaces of semistable Hitchin pairs on a reducible projective curve [Formula: see text]. When [Formula: see text] is the normalization of [Formula: see text], we give a birational morphism [Formula: see text] from the moduli space [Formula: see text] of good GPH on [Formula: see text] to the moduli space [Formula: see text] of Hitchin pairs on [Formula: see text] and show that the Hitchin map on [Formula: see text] induces a proper Hitchin map on [Formula: see text]. We determine the fibers of the Hitchin maps. We study the relationship between representations of the (topological) fundamental group of [Formula: see text] and Higgs bundles on [Formula: see text]. We show that if all the irreducible components of [Formula: see text] are smooth, then the Hitchin map is defined on the entire moduli space [Formula: see text].

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.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals EXTREMAL CASES OF RAPOPORT–ZINK SPACES

2021 ◽  
pp. 1-56
Author(s):  
Ulrich Görtz ◽  
Xuhua He ◽  
Michael Rapoport
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete List ◽  
Qualitative Properties ◽  
Model Case ◽  
Divisible Groups

Abstract We investigate qualitative properties of the underlying scheme of Rapoport–Zink formal moduli spaces of p-divisible groups (resp., shtukas). We single out those cases where the dimension of this underlying scheme is zero (resp., those where the dimension is the maximal possible). The model case for the first alternative is the Lubin–Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

Sign in / Sign up

Export Citation Format

Share Document