Positivity of Divisor Classes on the Strata of Differentials

Chen, Fan

doi:10.1093/qmath/haz032

“…The intersections with λ and the δ i are classical [CU, FP]. The degree of η can be computed by intersecting the relation ω ⊗k X /P 1 = π * η⊗O X (Σ 1 +• • •+Σ k ) valid on X with the class of one of the 2g − 2 exceptional divisors (this is as in [Che2], see also [Ghe,Example 3.2] for a similar computation).…”

Section: Test Curvesmentioning

confidence: 99%

$k$-canonical divisors through Brill-Noether special points

Gheorghita¹,

Tarasca²

2021

Preprint

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Inside the projectivized k-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence geometry and restrictions to various families, including pencils of curves on K3 surfaces and pencils of Du Val curves. We also show the extremality and rigidity of the closure of the incidence divisor consisting of smooth pointed curves together with a canonical or 2-canonical divisor passing through the marked point. MSC2020. 14H10, 14H51, 14C25 (primary), 30F30 (secondary).

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“…where ω π is the relative dualizing line bundle, each P i ⊂ E is the section corresponding to the marked point p i , Z ⊂ E is the closure of the locus of the zeros z i , and V ⊂ E is the vertical vanishing divisor arising from components of reducible curves E on which q is identically zero (see [4,Section 3]). Intersecting both sides of (4) with P 1 and applying π * , we conclude that…”

Section: Lemma 31 the Following Relation Of Divisor Classes Holds On Pqfmentioning

confidence: 99%

Dynamical invariants and intersection theory on the flex and gothic loci

Chen

¹

2021

European Journal of Mathematics

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2

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The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new SL 2 (R)-invariant varieties in Teichmüller dynamics, as discovered by McMullen-Mukamel-Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen-Möller-Sauvaget about computing sums of Lyapunov exponents for SL 2 (R)-invariant varieties via intersection theory.

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“…where ω π is the relative dualizing line bundle, each P i ⊂ E is the section corresponding to the marked point p i , Z ⊂ E is the closure of the locus of the zeros z i , and V ⊂ E is the vertical vanishing divisor arising from components of reducible curves E on which q is identically zero (see [C,Section 3]). Intersecting both sides of (4) with P 1 and applying π * , we conclude that…”

Section: Intersection Theory On the Gothic Locusmentioning

confidence: 99%

Dynamical invariants and intersection theory on the flex and gothic loci

Chen¹

2021

Preprint

Self Cite

0

View full text Add to dashboard Cite

The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new SL 2 (R)-invariant varieties in Teichmüller dynamics, as discovered by McMullen-Mukamel-Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen-Möller-Sauvaget about computing sums of Lyapunov exponents for SL 2 (R)-invariant varieties via intersection theory.

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Positivity of Divisor Classes on the Strata of Differentials

Abstract: Three decades ago Cornalba-Harris proved a fundamental positivity result for divisor classes associated to families of stable curves. In this paper we establish an analogous positivity result for divisor classes associated to families of stable differentials.

Cited by 4 publications

References 22 publications

$k$-canonical divisors through Brill-Noether special points

$k$-canonical divisors through Brill-Noether special points

Dynamical invariants and intersection theory on the flex and gothic loci

Dynamical invariants and intersection theory on the flex and gothic loci

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