Powers of the Theta Divisor and Relations in the Tautological Ring

Clader, Emily; Grushevsky, Samuel; Janda, Felix; Zakharov, Dmitry

doi:10.1093/imrn/rnx115

Cited by 10 publications

(14 citation statements)

References 35 publications

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“…A complete formula was conjectured by Pixton in 2014 and proven in [28]. For a sampling of the subsequent study and applications, see [6,14,19,20,25,26,38,48,49,50,51]. We refer the reader to [28, Section 0] and [42,Section 5] for more leisurely introductions to the subject.…”

Section: Double Ramification Cyclesmentioning

confidence: 99%

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Double ramification cycles with target varieties

Janda,

Pandharipande,

Pixton

et al. 2018

Preprint

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Let X be a nonsingular projective algebraic variety over C, and let M g,n,β (X) be the moduli space of stable maps f : (C, x 1 , . . . , x n ) → X from genus g, n-pointed curves C to X of degree β. Let S be a line bundle on X. Let A = (a 1 , . . . , a n ) be a vector of integers which satisfyConsider the following condition: the line bundle f * S has a meromorphic section with zeroes and poles exactly at the marked points x i with orders prescribed by the integers a i . In other words, we require f * S (− n i=1 a i x i ) to be the trivial line bundle on C. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X and is denoted by M ∼ g,A,β (X, S). The moduli space carries a virtual fundamental classin Gromov-Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push-forward via the forgetful morphism of [M ∼ g,A,β (X, S)] vir to M g,n,β (X). In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle proven in [28]. Several applications of the new formula are given.

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Section: Double Ramification Cyclesmentioning

confidence: 99%

“…For an unstable vertex of type (i) or (ii), we have both i ⊢v a i = 0 and β(v) = 0. The sum of the degrees of edges adjacent to such a vertex then vanishes by (14). However, the degree of every edge is a positive integer and the graph Φ is connected.…”

Section: Unstable Verticesmentioning

confidence: 99%

Double ramification cycles with target varieties

Janda,

Pandharipande,

Pixton

et al. 2018

Preprint

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“…Pixton's formula's opened new directions in the subject: new formulas for Hodge classes [35, Section 3], new relations in the tautological ring of

{\bar{scriptM}}_{g, n}

[17], new connections to the loci of meromorphic differentials [24, Appendix], and connections to new integrable hierarchies [9]. For a sampling of the subsequent study and applications, see [6–8, 10, 12, 16, 23, 31, 32, 45, 56, 57]. We refer the reader to [35, Section 0] and [50, Section 5] for more leisurely introductions to the subject.…”

Section: Introductionmentioning

confidence: 99%

Double ramification cycles with target varieties

Janda

Pandharipande

Pixton

et al. 2020

Journal of Topology

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Let X be a nonsingular projective algebraic variety over C, and let M g,n,β (X) be the moduli space of stable maps f : (C, x 1 ,. .. , x n) → X from genus g, n-pointed curves C to X of degree β. Let S be a line bundle on X. Let A = (a 1 ,. .. , a n) be a vector of integers which satisfy n i=1 a i = β c 1 (S). Consider the following condition: the line bundle f * S has a meromorphic section with zeroes and poles exactly at the marked points x i with orders prescribed by the integers a i. In other words, we require f * S (− n i=1 a i x i) to be the trivial line bundle on C. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X and is denoted by M ∼ g,A,β (X, S). The moduli space carries a virtual fundamental class [M ∼ g,A,β (X, S)] vir ∈ A * M ∼ g,A,β (X, S) in Gromov-Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push-forward via the forgetful morphism of [M ∼ g,A,β (X, S)] vir to M g,n,β (X). In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle proven in [28]. Several applications of the new formula are given. 1 Contents 0 Introduction 2 1 Curves with an rth root 13 2 GRR for the universal line bundle 27 3 Localization analysis 36 4 Applications 51 We define the set G g,n,β (X) of X-valued stable graphs as follows. A graph Γ ∈ G g,n,β (X) consists of the data

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“…Remark 5.8. The first part of the statement is very similar to [CGJZ16,Theorem 5], which gave a reduction algorithm based on Pixton's double ramification cycle. It confirms an expectation on page 7 of [BJP15], that "(.…”

Section: Tautological Ring Via Ppz Relationsmentioning

confidence: 79%

“…These graphs just correspond to the classes n i=1 ψ d i i κ e 1 ,...,e k for d i 0 and e i 1. There are still many relations among these classes that can be proved, in particular the relation R i (M g,n ) = 0 for i g; see [Loo95,Ion02] and also a recent new proof in [CGJZ16]. In the case i = g − 1, one can prove that dim R g−1 (M g,n ) n; see [Loo95,BSZ16] for the cases n = 1 and n 2, respectively.…”

Section: Introductionmentioning

confidence: 99%

The tautological ring of $\mathcal{M}_{g,n}$ via Pandharipande�Pixton�Zvonkine $r$-spin relations

Shadrin¹

2018

Alg. Geom.

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We use relations in the tautological ring of the moduli spaces M g,n derived by Pandharipande, Pixton, and Zvonkine from the Givental formula for the r-spin Witten class in order to obtain some restrictions on the dimensions of the tautological rings of the open moduli spaces M g,n. In particular, we give a new proof for the result of Looijenga (for n = 1) and Buryak et al. (for n 2) that dim R g−1 (M g,n) n. We also give a new proof of the result of Looijenga (for n = 1) and Ionel (for arbitrary n 1) that R i (M g,n) = 0 for i g and give some estimates for the dimension of R i (M g,n) for i g − 2.

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Powers of the Theta Divisor and Relations in the Tautological Ring

Cited by 10 publications

References 35 publications

Double ramification cycles with target varieties

Double ramification cycles with target varieties

Double ramification cycles with target varieties

The tautological ring of $\mathcal{M}_{g,n}$ via Pandharipande�Pixton�Zvonkine $r$-spin relations

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