Projectivity of the moduli space of stable curves

2011 ◽  
pp. 399-439
Author(s):  
Enrico Arbarello ◽  
Maurizio Cornalba ◽  
Phillip A. Griffiths
Keyword(s):  
Moduli Space ◽  
Stable Curves

scholarly journals Enumerating Hassett’s Wall and Chamber Decomposition of the Moduli Space of Weighted Stable Curves

2018 ◽  
Vol 29 (1) ◽  
pp. 36-53
Author(s):  
Kenneth Ascher ◽  
Connor Dubé ◽  
Daniel Gershenson ◽  
Elaine Hou
Keyword(s):  
Moduli Space ◽  
Stable Curves

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 Numerical criteria for divisors on to be ample

2009 ◽  
Vol 145 (5) ◽  
pp. 1227-1248 ◽  
Author(s):  
Angela Gibney
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Topological Type ◽  
Rational Curves ◽  
Canonical Divisor ◽  
Content Type ◽  
One Dimensional ◽  
Stable Curves ◽  
The One ◽  
Ample Divisors

AbstractThe moduli space $\M _{g,n}$ of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k. The one-dimensional components of this stratification are smooth rational curves called F-curves. These are believed to determine all ample divisors. F-conjecture A divisor on $\M _{g,n}$ is ample if and only if it positively intersects theF-curves. In this paper, proving the F-conjecture on $\M _{g,n}$ is reduced to showing that certain divisors on $\M _{0,N}$ for N⩽g+n are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. Numerical criteria and an algorithm are given to check whether a divisor is ample. By using a computer program called the Nef Wizard, written by Daniel Krashen, one can verify the conjecture for low genus. This is done on $\M _g$ for g⩽24, more than doubling the number of cases for which the conjecture is known to hold and showing that it is true for the first genera such that $\M _g$ is known to be of general type.

scholarly journals Fibonacci, Golden Ratio, and Vector Bundles

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 426
Author(s):  
Noah Giansiracusa
Keyword(s):  
Lie Algebra ◽  
Moduli Space ◽  
Vector Bundles ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Theoretical Physics ◽  
Stable Curves ◽  
Exceptional Lie Algebra ◽  
Summation Formulas

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

scholarly journals Brill–Noether loci in codimension two

2013 ◽  
Vol 149 (9) ◽  
pp. 1535-1568 ◽  
Author(s):  
Nicola Tarasca
Keyword(s):  
Moduli Space ◽  
Moduli Space Of Curves ◽  
Stable Curves ◽  
Codimension Two

AbstractLet us consider the locus in the moduli space of curves of genus$2k$defined by curves with a pencil of degree$k$. Since the Brill–Noether number is equal to$- 2$, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.

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 EXTENDING THE DOUBLE RAMIFICATION CYCLE BY RESOLVING THE ABEL-JACOBI MAP

2019 ◽  
pp. 1-29 ◽  
Author(s):  
David Holmes
Keyword(s):  
Moduli Space ◽  
Smooth Curves ◽  
Fundamental Class ◽  
Stable Curves ◽  
Double Ramification Cycle

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel–Jacobi map. This breaks down over the boundary since the Abel–Jacobi map fails to extend. We construct a ‘universal’ resolution of the Abel–Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.

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