Divisor classes associated to families of stable varieties, with applications to the moduli space of curves

Cornalba, Maurizio; Harris, Joe

doi:10.24033/asens.1564

Cited by 170 publications

(252 citation statements)

References 12 publications

Supporting

Mentioning

248

Contrasting

Unclassified

Order By: Relevance

“…The reader can consult [8] and [9] for details of the discussion here, but we give a comment here: if S is regular of dimension 1 and if G is a finitely supported O S -module, then Div(G) is an effective divisor such that for each prime divisor s ∈ S, the coefficient of s is equal to the length of G at s. 2 There exists the corresponding morphism Y → Aut Mg Zg . In general, let X be a noetherian algebraic stack, Y a closed substack of X , and let h : T → X be a morphism from a scheme T .…”

Section: The Statement and An Applicationmentioning

confidence: 99%

“…Therefore, we could define ξ j 's as a divisor class on I g,C without being nervous (c.f. [2]). In our case, however, we do not have enough information on the geometry of I g in characteristic 2 and cannot easily defined them as the class of locuses.…”

Section: Boundary Classesmentioning

confidence: 99%

“…Let δ i (X/Y ) denote the number of the nodes of type i, ξ 0 (X/Y ) the number of nodes of subtype 0 and let ξ j (X/Y ) denote the number of pairs of nodes {x, ι(x)} of subtype j > 0, in all the fibers. Cornalba and Harris proved in [2] an equality…”

mentioning

confidence: 99%

“…Before dealing with the equality in positive characteristic, let us recall the proof over C in [2]. Let I g,C be the moduli of stable hyperelliptic curves of genus g over C and I g,C the open dense subset consisting of smooth hyperelliptic curves.…”

mentioning

confidence: 99%

“…Then for each node x of type 0 in a fiber, we can assign a non-negative integer, called the subtype, to x or the pair {x, ι(x)} (c.f. [2] or § § 1.2 for the definitions). Let δ i (X/Y ) denote the number of the nodes of type i, ξ 0 (X/Y ) the number of nodes of subtype 0 and let ξ j (X/Y ) denote the number of pairs of nodes {x, ι(x)} of subtype j > 0, in all the fibers.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic

Yamaki¹

2004

Asian Journal of Mathematics

View full text Add to dashboard Cite

Section: The Statement and An Applicationmentioning

confidence: 99%

Section: Boundary Classesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic

Yamaki¹

2004

Asian Journal of Mathematics

View full text Add to dashboard Cite

Slope of Cusp Forms and Theta Series

Manni¹

2000

Journal of Number Theory

View full text Add to dashboard Cite

The aim of this paper is to improve the results of [o] about the theta series associated to the even extremal unimodular quadratic form. We prove that, in degree 3, the associated theta series is still unique for extremal even unimodular lattices of rank 32 and 48. For the ranks 40, 56, 72 we analyze the situation. We discuss also consequences in degree 4. To obtain these results we use some known results about the vanishing of the cusp forms on the hyperelliptic locus I g . We give also some improvements in this subject, namely we show that the estimate about the slope of cusp forms vanishing on I g is sharp.

show abstract

Classical and minimal models of the moduli space of curves of genus two

Hassett

Progress in Mathematics

View full text Add to dashboard Cite

1 space of stable curves is its own log canonical model (see Theorem 4.7). Our basic strategy it to interpolate between the log canonical model and the (conjectural) canonical model by consideringThis program is a subject of ongoing work, inspired by correspondence with S. Keel, in collaboration with D. Hyeon. Future papers will address the stable behavior of these spaces for successively smaller values of α. It is remarkable that their behavior is largely independent of the genus (see, for example, Remark 4.9.)However, for small values of g special complexities arise. When g = 2 or 3, the locus in M g of curves with automorphism has codimension ≤ 1. To include these spaces under our general framework, we must take into account the properties of the moduli stack M g . In particular, it is necessary to use the canonical divisor of the moduli stack rather than its coarse moduli space. These differ substantially, as the natural morphism M g → M g is ramified at stable curves admitting automorphisms.Luckily, we have inherited a tremendously rich literature on curves of small genus. The invariant-theoretic properties of M 2 were extensively studied by the 19th century German school [Cl], who realized it as an open subset of the weighted projective space P(1, 2, 3, 5). Theorem 4.10 reinterprets this classical construction using the modern language of stacks and minimal models.We work over an algebraically closed field k of characteristic zero. We use the notation ≡ for Q-linear equivalence of divisors. Throughout, a curve is a connected, projective, reduced scheme of dimension one. The genus of a curve is its arithmetic genus.The moduli stack of smooth (resp. stable) curves of genus g is denoted M g (resp. M g ); the corresponding coarse moduli scheme is denoted M g (resp. M g ). The boundary divisors in M g (resp. M g ) are denoted ∆ 0 , ∆ 1 , . . . , ∆ ⌊g/2⌋ (resp. δ 0 , δ 1 , . . . , δ ⌊g/2⌋ ).Let Q : M g → M g denote the natural morphism from the moduli stack to the coarse moduli space, so that Q * ∆ i = δ i , i = 1 Q * ∆ 1 = 2δ 1 .

show abstract

Divisor classes associated to families of stable varieties, with applications to the moduli space of curves

Cited by 170 publications

References 12 publications

Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic

Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic

Slope of Cusp Forms and Theta Series

Classical and minimal models of the moduli space of curves of genus two

Contact Info

Product

Resources

About