scholarly journals Fundamental groups of moduli stacks of stable curves of compact type

2009 ◽  
Vol 13 (1) ◽  
pp. 247-276 ◽  
Author(s):  
Marco Boggi
Keyword(s):  
Fundamental Groups ◽  
Compact Type ◽  
Stable Curves ◽  
Moduli Stacks

scholarly journals Normal functions and the height of Gross-Schoen cycles

2014 ◽  
Vol 214 ◽  
pp. 53-77 ◽  
Author(s):  
Robin De Jong
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
The Self ◽  
Compact Type ◽  
Stable Curves ◽  
Normal Functions ◽  
Chern Form ◽  
Bloch Line ◽  
Pointed Curve ◽  
Dualizing Sheaf

AbstractWe prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

scholarly journals Normal functions and the height of Gross-Schoen cycles

2014 ◽  
Vol 214 ◽  
pp. 53-77
Author(s):  
Robin De Jong
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
The Self ◽  
Compact Type ◽  
Stable Curves ◽  
Normal Functions ◽  
Chern Form ◽  
Bloch Line ◽  
Pointed Curve ◽  
Dualizing Sheaf

AbstractWe prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

Second flip in the Hassett–Keel program: existence of good moduli spaces

2017 ◽  
Vol 153 (8) ◽  
pp. 1584-1609 ◽  
Author(s):  
Jarod Alper ◽  
Maksym Fedorchuk ◽  
David Ishii Smyth
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
General Criterion ◽  
Stable Curves ◽  
Algebraic Stack ◽  
Moduli Stacks ◽  
Local Description ◽  
Compositio Math

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel–Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne–Mumford stack. We apply this result to prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$-stable curves introduced in [J. Alper et al., Second flip in the Hassett–Keel program: a local description, Compositio Math. 153 (2017), 1547–1583] admit good moduli spaces.

scholarly journals Socle pairings on tautological rings

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Felix Janda ◽  
Aaron Pixton
Keyword(s):  
Explicit Formula ◽  
Moduli Space ◽  
Compact Type ◽  
Stable Curves ◽  
Tautological Ring ◽  
Tautological Rings

We study some aspects of the $\lambda_g$ pairing on the tautological ring of $M_g^c$, the moduli space of genus $g$ stable curves of compact type. We consider pairing kappa classes with pure boundary strata, all tautological classes supported on the boundary, or the full tautological ring. We prove that the rank of this restricted pairing is equal in the first two cases and has an explicit formula in terms of partitions, while in the last case the rank increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on the tautological ring of $M_g$. Comment: 18 pages, 1 figure; v3: journal version; v2: minor revisions to sections 1.1 and 4.1, results unchanged

scholarly journals Moduli Stacks of Permutation Classes of Pointed Stable Curves

2008 ◽  
Vol 76 (1) ◽  
pp. 401-418
Author(s):  
Jörg Zintl
Keyword(s):  
Stable Curves ◽  
Moduli Stacks

scholarly journals $$\pi _1$$ of Miranda moduli spaces of elliptic surfaces

2021 ◽  
Author(s):  
Michael Lönne
Keyword(s):  
Fundamental Group ◽  
Moduli Spaces ◽  
Classical Result ◽  
Projective Line ◽  
The Other ◽  
Fundamental Groups ◽  
Elliptic Surfaces ◽  
Generators And Relations ◽  
Moduli Stacks ◽  
Finite Presentations

AbstractWe give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over $${\mathbf {P}}^1$$ P 1 which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of $$SL_2{{\mathbb {Z}}}$$ S L 2 Z presented in terms of $$\Bigg (\begin{array}{ll} 1&{}1\\ 0&{}1\end{array} \Bigg )$$ ( 1 1 0 1 ) , $$\Bigg (\begin{array}{ll} 1&{}0\\ {-1}&{}1\end{array} \Bigg )$$ ( 1 0 - 1 1 ) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other.Our approach exploits the natural $${\mathbb {Z}}_2$$ Z 2 -action on Weierstrass curves and the identification of $${\mathbb {Z}}_2$$ Z 2 -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over $${{\mathbf {P}}}^1$$ P 1 . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition.

scholarly journals Modular compactifications of the space of pointed elliptic curves I

2010 ◽  
Vol 147 (3) ◽  
pp. 877-913 ◽  
Author(s):  
David Ishii Smyth
Keyword(s):  
Stability Conditions ◽  
Model Program ◽  
Arithmetic Genus ◽  
Minimal Model Program ◽  
Stable Curves ◽  
Moduli Stacks ◽  
Log Minimal Model Program ◽  
Curve Singularities ◽  
Definition Of ◽  
Genus One

AbstractWe introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne–Mumford stability. For every pair of integers 1≤m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne–Mumford stack $\overline {\mathcal {M}}_{1,n}(m)$. We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks $\overline {\mathcal {M}}_{1,\mathcal {A}}(m)$. In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for $\overline {M}_{1,n}$.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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