Genus-one stable maps, local equations, and Vakil–Zinger’s desingularization

Hu, Yi; Li, Jun

doi:10.1007/s00208-010-0504-8

Cited by 29 publications

(111 citation statements)

References 10 publications

Supporting

Mentioning

109

Contrasting

Order By: Relevance

“…The minimal genus one connected component may correspond to a cycle in a dual curve, then contract it to the unique vertex which will be the root. Finally, to make this tree to be terminally weighted, we do some pruning process; see [7,Section 3.4] for the pruning process.…”

Section: Notationsmentioning

confidence: 99%

“…The morphism b is the same as a desingularization constructed by Vakil and Zinger in [15]. In [7], Hu and Li found local defining equation for k " 0. We note that in [7,Remark 5.7], Hu and Li mentioned that their results can be extended when the marked points exist.…”

Section: Introductionmentioning

confidence: 97%

“…In [7], Hu and Li found local defining equation for k " 0. We note that in [7,Remark 5.7], Hu and Li mentioned that their results can be extended when the marked points exist. For a desingularization b defined by Vakil and Zinger, the condition (1) above is trivial and (2) is satisfied by [1,Proposition 7.2].…”

Section: Introductionmentioning

confidence: 99%

“…For a desingularization b defined by Vakil and Zinger, the condition (1) above is trivial and (2) is satisfied by [1,Proposition 7.2]. In Section 2, we will provide a detail about conditions (3) and (4) mentioned in the last paragraph of [7].…”

Section: Introductionmentioning

confidence: 99%

“…be the universal curve and ev : C Ñ P n be the evaluation morphism. It is known that for l ą 0, π˚ev˚O P n plq is locally free by [7,Theorem 2.11]. Let N red :" π˚ev˚p' m i"1 O P n pdegf i qq| Ă By definition, when α is a pullback class from pP n q k we note that GW red 1,d pαq is equal to the degree of a class epN red q X r Ă…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Algebraic Reduced Genus One Gromov–Witten Invariants for Complete Intersections in Projective Spaces

Lee

Oh²

2019

International Mathematics Research Notices

View full text Add to dashboard Cite

In [17,18], Zinger defined reduced Gromov-Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). In [3], Chang and Li provided a proof of the comparison theorem for quintic Calabi-Yau 3-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. In [5], Coates and Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger in [14] and proved the comparison theorem for every Calabi-Yau threefold. In this paper, we prove the comparison theorem for every (not necessarily Calabi-Yau) complete intersection of dimension 2 or 3 in projective spaces by taking a definition of reduced GW invariants in [5]. GW 1,d´G W red 1,d " 1 12 GW 0,d . (1.1) In [5], Coates and Manolache proved this result for every smooth projective Calabi-Yau 3-fold Q; using their definition of reduced GW invariants. M red Q,k,d be a vector bundle on a space Ă M red Q,k,d :" Ă M red k,dˆM 1,k pP n ,dq red M 1,k pQ, dq red . It is proven that the normal cone C Ă M red Q,k,d { Ă M red k,d is embedded in a total space of N red in [5]. Definition 1.2 ([5]). The virtual fundamental class rM red 1,k pQ, dqs vir is defined by rM red 1,k pQ, dqs vir :" b 1˚p 0 ! N red rC Ă M red Q,k,d { Ă M red k,d sq P A˚pM red 1,k pQ, dqq Q where b 1 : Ă M red Q,k,d Ñ M red 1,k pQ, dq is a restriction of b.

show abstract

Section: Notationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Algebraic Reduced Genus One Gromov–Witten Invariants for Complete Intersections in Projective Spaces

Lee

Oh²

2019

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

The geometry of stable quotients in genus one

Cooper

2014

Math. Ann.

View full text Add to dashboard Cite

Stable quotient spaces provide an alternative to stable maps for compactifying spaces of maps. When n ≥ 2, the space) compactifies the space of degree d maps of smooth genus g curves to P n−1 , whileis a quotient of a Hassett weighted pointed space. In this paper we study the coarse moduli schemes associated to the smooth proper Deligne-Mumford stacks Q 1 (P n−1 , d), for all n ≥ 1. We show these schemes are projective, rationally connected and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, and the cones of ample and effective divisors. We conclude that Q 1 (P n−1 , d) is Fano if and only if n(d − 1)(d + 2) < 20. In the case n = 1, we write in addition a closed formula for the Poincaré polynomial.

show abstract

Quantum Lefschetz property for genus two stable quasimap invariants

Lee,

Li,

2023

Math. Ann.

View full text Add to dashboard Cite

By the reduced component in a moduli space of stable quasimaps to n-dimensional projective space $$\mathbb {P}^n$$ P n we mean the closure of the locus in which the domain curves are smooth. As in the moduli space of stable maps, we prove the reduced component is smooth in genus 2, degree$$\ge 3$$ ≥ 3 . Then we prove the virtual fundamental cycle of the moduli space of stable quasimaps to a complete intersectionXin $$\mathbb {P}^n$$ P n of genus 2, degree $$\ge 3$$ ≥ 3 is explicitly expressed in terms of the fundamental cycle of the reduced component of $$\mathbb {P}^n$$ P n and virtual cycles of lower genus $$<2$$ < 2 moduli spaces of X.

show abstract

Genus-one stable maps, local equations, and Vakil–Zinger’s desingularization

Cited by 29 publications

References 10 publications

Algebraic Reduced Genus One Gromov–Witten Invariants for Complete Intersections in Projective Spaces

Algebraic Reduced Genus One Gromov–Witten Invariants for Complete Intersections in Projective Spaces

The geometry of stable quotients in genus one

Quantum Lefschetz property for genus two stable quasimap invariants

Contact Info

Product

Resources

About