A desingularization of the main component of the moduli space of genus-one stable maps into ℙn

By Ravi Vakil, A. Zinger. 2006

71 citations recorded by the source corpus.

We construct a natural smooth compactification of the space of smooth genus-one curves with k distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps S M0;k.P n ;d/. In fact, our compactification is obtained from the singular space of stable genus-one maps S M1;k.P n ;d/ through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component S M 0;k .P n ;d/ of S M1;k.P n ;d/. A number of applications of these desingularizations in enumerative geometry and Gromov‐Witten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus-one Gromov‐Witten invariants of a quintic threefold. 14D20; 53D99

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0e2f4b93ce05d04a14c5e8d1d5d742e96f161f48

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Citations in the corpus

Standard versus reduced genus-one Gromov–Witten invariants (2007, methodology): “tion for genus-one invariants is addressed in [Z3] and [LZ] by deﬁning reducedgenus-one GW-invariants and showing that these invariants do satisfy the hyperplane relation, respectively. Combined with [VaZ], this last approach provides a platform suitable for an application of combinatorial technique”
Logarithmic Stable Maps (2008, methodology): “tack. Hence it provides a moduli-theoretic desingularization of the main component of stable maps. It would be interesting to ﬁnd an explicit relationship with Vakil and Zinger’s desingularization in [33]. The space Mlog,ch 1 (X+/X) can be perhaps used to algebro-geometrically establish quantum Lefs”
The reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces (2007, methodology): “es is Z Mf 1,|val(v0)| ψ˜|val(v0)| = (|val(v 0)|−1)! 24 if k=1,14 (1.34) where ψ˜=c 1(L∗) is the universal ψ-class; see [Z3, Corollary 1.2]. The ﬁnal piece of localization data we need to recall from [VaZ] is that e(V′ 1) Ze Γ = π∗ B V′ Γ;B/π ∗ 1(L ∗ Γ⊗L µ(v0))⊗π ∗ 2γ ∗ , (1.35) where V′ Γ;B= Y e∈Ed”
The moduli space of stable quotients (2009, background): “P n (1); d/ in the moduli of stable maps (see Hu and Li [15] and Vakil and Zinger [43]), the stable quotient space achieves a simple modular desingularization by blowing down.”