scholarly journals A compactification of the moduli space of self-maps of $\mathbb {CP}^1$ via stable maps

2017 ◽  
Vol 21 (11) ◽  
pp. 273-318
Author(s):  
Johannes Schmitt
Keyword(s):  
Moduli Space ◽  
Stable Maps

scholarly journals Towards Mori’s program for the moduli space of stable maps

2011 ◽  
Vol 133 (5) ◽  
pp. 1389-1419 ◽  
Author(s):  
Dawei Chen ◽  
Izzet Coskun
Keyword(s):  
Moduli Space ◽  
Stable Maps

The Effective Cone of the Kontsevich Moduli Space

2008 ◽  
Vol 51 (4) ◽  
pp. 519-534 ◽  
Author(s):  
Izzet Coskun ◽  
Joe Harris ◽  
Jason Starr
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete Characterization ◽  
Stable Maps ◽  
Linear Combinations ◽  
Effective Divisors ◽  
Effective Cone

AbstractIn this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, , stabilize when r ≥ d. We give a complete characterization of the effective divisors on . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

scholarly journals MODULI SPACE OF STABLE MAPS TO PROJECTIVE SPACE VIA GIT

2010 ◽  
Vol 21 (05) ◽  
pp. 639-664 ◽  
Author(s):  
YOUNG-HOON KIEM ◽  
HAN-BOM MOON
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Blow Up ◽  
Stable Maps ◽  
Homogeneous Polynomials ◽  
Birational Map

We compare the Kontsevich moduli space [Formula: see text] of stable maps to projective space with the quasi-map space ℙ( Sym d(ℂ2) ⊗ ℂn)//SL(2). Consider the birational map [Formula: see text] which assigns to an n tuple of degree d homogeneous polynomials f1, …, fn in two variables, the map f = (f1 : ⋯ : fn) : ℙ1 → ℙn-1. In this paper, for d = 3, we prove that [Formula: see text] is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces [Formula: see text] with d = 1, 2. In particular, [Formula: see text] is the SL(2)-quotient of a smooth rational projective variety. The degree two case [Formula: see text], which is the blow-up of ℙ( Sym 2ℂ2 ⊗ ℂn)//SL(2) along ℙn-1, is worked out as a preliminary example.

scholarly journals The moduli space of regular stable maps

2007 ◽  
Vol 259 (3) ◽  
pp. 525-574 ◽  
Author(s):  
Joel W. Robbin ◽  
Yongbin Ruan ◽  
Dietmar A. Salamon
Keyword(s):  
Moduli Space ◽  
Stable Maps

scholarly journals Stable maps and stable quotients

2014 ◽  
Vol 150 (9) ◽  
pp. 1457-1481 ◽  
Author(s):  
Cristina Manolache
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Stable Maps ◽  
Intersection Numbers ◽  
Fundamental Class ◽  
Push Forward ◽  
The Relationship

AbstractWe analyze the relationship between two compactifications of the moduli space of maps from curves to a Grassmannian: the Kontsevich moduli space of stable maps and the Marian–Oprea–Pandharipande moduli space of stable quotients. We construct a moduli space which dominates both the moduli space of stable maps to a Grassmannian and the moduli space of stable quotients, and equip our moduli space with a virtual fundamental class. We relate the virtual fundamental classes of all three moduli spaces using the virtual push-forward formula. This gives a new proof of a theorem of Marian–Oprea–Pandharipande: that enumerative invariants defined as intersection numbers in the stable quotient moduli space coincide with Gromov–Witten invariants.

scholarly journals Quantum cohomology of [ℂN/μr]

2010 ◽  
Vol 146 (5) ◽  
pp. 1291-1322 ◽  
Author(s):  
Arend Bayer ◽  
Charles Cadman
Keyword(s):  
Cyclic Group ◽  
Moduli Space ◽  
Chern Class ◽  
Quantum Cohomology ◽  
Closed Formula ◽  
Stable Maps ◽  
Genus Zero ◽  
Witten Theory

AbstractWe give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of rth root constructions on $\overline {M}_{0, n}$. We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

scholarly journals Sheaves of maximal intersection and multiplicities of stable log maps

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Jinwon Choi ◽  
Michel van Garrel ◽  
Sheldon Katz ◽  
Nobuyoshi Takahashi
Keyword(s):  
Moduli Space ◽  
General Position ◽  
Deformation Theory ◽  
Duke Math ◽  
Stable Maps ◽  
Natural Components ◽  
Theoretical Results

AbstractA great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.

The Ample Cone of the Kontsevich Moduli Space

2009 ◽  
Vol 61 (1) ◽  
pp. 109-123 ◽  
Author(s):  
Izzet Coskun ◽  
Joe Harris ◽  
Jason Starr
Keyword(s):  
Moduli Space ◽  
Stable Maps ◽  
Ample Cone ◽  
Free Divisors

Abstract.We produce ample (resp. NEF, eventually free) divisors in the Kontsevich space of n-pointed, genus 0, stable maps to ℙr, given such divisors in We prove that this produces all ample (resp. NEF, eventually free) divisors in As a consequence, we construct a contraction of the boundary analogous to a contraction of the boundary first constructed by Keel and McKernan.

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