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 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

scholarly journals NEW EXAMPLES OF CALABI–YAU 3-FOLDS AND GENUS ZERO SURFACES

2014 ◽  
Vol 16 (02) ◽  
pp. 1350010 ◽  
Author(s):  
GILBERTO BINI ◽  
FILIPPO F. FAVALE ◽  
JORGE NEVES ◽  
ROBERTO PIGNATELLI
Keyword(s):  
Fundamental Group ◽  
Moduli Space ◽  
General Type ◽  
Picard Number ◽  
Surfaces Of General Type ◽  
Geometric Genus ◽  
Genus Zero ◽  
New Family ◽  
Hodge Numbers ◽  
Projective Lines

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

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 Relative Quasimaps and Mirror Formulae

2020 ◽  
Author(s):  
Luca Battistella ◽  
Navid Nabijou
Keyword(s):  
Generating Function ◽  
Moduli Space ◽  
Toric Variety ◽  
Recursion Formula ◽  
Genus Zero ◽  
Relative Invariant ◽  
Lefschetz Theorem ◽  
Virtual Class

Abstract We construct and study the theory of relative quasimaps in genus zero, in the spirit of Gathmann. When $X$ is a smooth toric variety and $Y$ is a smooth very ample hypersurface in $X$, we produce a virtual class on the moduli space of relative quasimaps to $(X,Y)$, which we use to define relative quasimap invariants. We obtain a recursion formula which expresses each relative invariant in terms of invariants of lower tangency, and apply this formula to derive a quantum Lefschetz theorem for quasimaps, expressing the restricted quasimap invariants of $Y$ in terms of those of $X$. Finally, we show that the relative $I$-function of Fan–Tseng–You coincides with a natural generating function for relative quasimap invariants, providing mirror-symmetric motivation for the theory.

scholarly journals SUPERPOTENTIAL OF THE M-THEORY CONIFOLD AND TYPE IIA STRING THEORY

2004 ◽  
Vol 19 (04) ◽  
pp. 521-555 ◽  
Author(s):  
GOTTFRIED CURIO
Keyword(s):  
String Theory ◽  
Heisenberg Group ◽  
Group Action ◽  
Moduli Space ◽  
Monodromy Group ◽  
Witten Theory ◽  
Flat Bundle ◽  
M Theory

The membrane instanton superpotential for M-theory on the G2 holonomy manifold given by the cone on S3×S3 is given by the dilogarithm and has Heisenberg monodromy group in the quantum moduli space. We compare this to a Heisenberg group action on the type IIA hypermultiplet moduli space for the universal hypermultiplet, to metric corrections from membrane instantons related to a twisted dilogarithm for the deformed conifold and to a flat bundle related to a conifold period, the Heisenberg group and the dilogarithm appearing in five-dimensional Seiberg/Witten theory.

ON GROMOV–WITTEN INVARIANTS OF DEL PEZZO SURFACES

2013 ◽  
Vol 24 (07) ◽  
pp. 1350054 ◽  
Author(s):  
MENDY SHOVAL ◽  
EUGENII SHUSTIN
Keyword(s):  
Quantum Cohomology ◽  
Enumerative Geometry ◽  
Del Pezzo Surfaces ◽  
Manuscripta Math ◽  
Moduli Space Of Curves ◽  
Divisor Class ◽  
Genus Zero ◽  
Type Formula ◽  
Del Pezzo ◽  
Generic Points

We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81–148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ2 and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math.102 (2000) 53–84]. We solve the problem in two steps: (1) we consider curves on [Formula: see text], the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso–Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type [Formula: see text], the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov–Witten invariants of [Formula: see text] via enumerative invariants of [Formula: see text], using Vakil's extension of the Abramovich–Bertram formula.

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.

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