A desingularization problem in the theory of Siegel modular functions

Igusa, Jun-ichi

doi:10.1007/bf01361555

Cited by 70 publications

(72 citation statements)

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“…For every ζ e 3> u e GL(g, Z)(n) and τ' e K λ write ζ ι u in the form ζ o + τ'm + n with ζ o eZ and with m, n = 0 mod. n. Put Re (L τ ,(ζ, ζ)) = Ίm ζ Im (τO" 1 Im ζ after Igusa [8]. Then by (2.4.3) we have…”

Section: (Cf [8] Lemma 9) Let σ Be a Half-integer Positive Matrix Imentioning

confidence: 95%

“…In fact we show the existence of j first and we show that j can be lifted to j. In the proof we use the methods introduced by Igusa in [8] in its full extent. Especially we use the notions of Fourier-Jacobi series and central cones.…”

mentioning

confidence: 88%

“…The latter is defined as follows. Let v be an element of $)g such that the minimum of values *QVQ for g e Z 9 is equal to 1 and that g *g with such integral vectors g as 1 In this section we make a review on Satake compactifications and their monoidal transforms along the boundary which were introduced by Igusa [8]. For details that we omitted to prove here, we refer the reader to [8].…”

Section: This Representation Being Injective We Identify π G+1 With mentioning

confidence: 99%

“…of the Satake compactiίication by blowing-up along the boundary ( [8]). Unfortunately this procedure does not give the desingularization if g > 3.…”

mentioning

confidence: 99%

“…In Section 1 we recall the notions of fundamental cones and central cones in the theory of positive symmetric matrices due to Igusa [8] and we prove Theorem 1. In Section 2 we recall Satake compactifications and introduce the Igusa monoidal transforms with Igusa's fundamental results in [8]. In Section 3 we make a brief summary on the theory of stable curves due to Deligne and Mumford ( [4]).…”

mentioning

confidence: 99%

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On the canonical holomorphic map from the Moduli space of stable curves to the Igusa monoidal transform

Namikawa

1973

Nagoya Mathematical Journal

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Section: (Cf [8] Lemma 9) Let σ Be a Half-integer Positive Matrix Imentioning

confidence: 95%

mentioning

confidence: 88%

Section: This Representation Being Injective We Identify π G+1 With mentioning

confidence: 99%

“…of the Satake compactiίication by blowing-up along the boundary ( [8]). Unfortunately this procedure does not give the desingularization if g > 3.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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On the canonical holomorphic map from the Moduli space of stable curves to the Igusa monoidal transform

Namikawa

1973

Nagoya Mathematical Journal

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

Hassett

Progress in Mathematics

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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 .

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Moduli of Abelian Varieties

Chai

1991

Number Theory

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A desingularization problem in the theory of Siegel modular functions

Cited by 70 publications

References 15 publications

On the canonical holomorphic map from the Moduli space of stable curves to the Igusa monoidal transform

On the canonical holomorphic map from the Moduli space of stable curves to the Igusa monoidal transform

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

Moduli of Abelian Varieties

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