On the Birational Geometry of the Universal Picard Variety

Abstract

We compute the Kodaira dimension of the universal Picard variety P d,g parameterizing line bundles of degree d on curves of genus g under the assumption that (d − g + 1, 2g − 2) = 1. We also give partial results for arbitrary degrees d and we investigate for which degrees the universal Picard varieties are birational.1991 Mathematics Subject Classification. 14H10. -Geometria delle varietà algebriche e dei loro spazi di moduli. 1 2 GILBERTO BINI, CLAUDIO FONTANARI, FILIPPO VIVIANI since P d,g is singular and not projective, the Kodaira dimension of P d,g , which we denote by κ(P d,g ), is defined as the Kodaira dimension of any smooth projective model of it (see [Laz04, Example 2.1.5]). The previous result of Verra implies that κ(P d,g ) = −∞ for g ≤ 9 and any d.Theorem 1.2. Assume that (d − g + 1, 2g − 2) = 1 and g ≥ 10. The Kodaira dimension of P d,g is equal toIn Propositions 6.5 and 6.3, we also determine the Iitaka fibration (see [Laz04, Def. 1.3.6]) of P d,g in the non-trivial cases, namely for g ≥ 11. Without any assumption on the degree d, we obtain the following partial result:Theorem 1.3. The Kodaira dimension of P d,g (for g ≥ 10) satisfies the following inequalitiesMoreover, κ(P d,g ) = 3g −3 if κ(M g ) ≥ 0 (and in particular for g ≥ 22).Let us now explain the strategy that we use to prove the above results. The main tool we use is the GIT compactification constructed by Caporaso (see [Cap94]) φ d : P d,g → M g of P d,g over the Deligne-Mumford moduli space M g of stable curves of genus g. The projective normal variety P d,g is a good moduli space for the stack Pic d,g (see [Cap08] and [Mel09]), whose section over a scheme S is the groupoid Pic d,g (S) of families of quasistable curves of genus g f : (C, L) → S endowed with a balanced line bundle L of degree d (see 2.1 for details). Furthermore, P d,g is a coarse moduli scheme for Pic d,g if and only if (d − g + 1, 2g − 2) = 1, which is precisely the numerical hypothesis on the degree d in Theorem 1.2.Albeit P d,g is singular, we can prove (under the same assumption on the degree) that P d,g has canonical singularities and therefore pluricanonical forms on the smooth locus lift to any desingularization:Theorem 1.4. Assume that (d−g+1, 2g−2) = 1 and that g ≥ 4. Then P d,g has canonical singularities. In particular, if P d,g is a resolution ON THE BIRATIONAL GEOMETRY OF P d,g 3 of singularities of P d,g , then every pluricanonical form defined on the smooth locus P reg d,g of P d,g extends holomorphically to P d,g , that is, for all integers m we have h 0 (P reg d,g , mK P reg d,g ) = h 0 ( P d,g , mK P d,g ). The proof of this theorem is given in Section 4. The restriction on the degree d comes from the fact that P d,g has finite quotient singularities if and only if (d − g + 1, 2g − 2) = 1; hence only for such degrees d we can apply the Reid-Tai criterion for the canonicity of finite quotient singularities (see e.g. [HM82, p. 27] or [Lud07, Thm. 4.1.11]). Indeed, we establish in Theorem 4.8 a similar statement without any restriction on d for the open ...

The local structure of compactified Jacobians

“…When X does not admit a non-trivial automorphism, the authors will prove this result by using the explicit description of the completed local ring in Theorem A, and in general, they will reduce the proof to a similar argument using a generalization of the Reid-Tai-Shepherd-Barron criterion for toric singularities. The results in [13] will extend the work of Bini, Fontanari and the third author [6], where it is shown thatJ d,g has canonical singularities when gcd(d + 1 − g, 2g − 2) = 1, a condition equivalent to the condition thatJ d,g has finite quotient singularities. Under the same assumption on d and g, the same authors computed the Kodaira dimension and the Itaka fibration ofJ d,g ([6, Theorem 1.2]), and in [13], the present authors will extend that computation to all d, g.…”

The singularities and birational geometry of the compactified universal Jacobian

“…We take the point (C, I) ∈J d,g , and consider its image C ∈ M g . Then we break the argument into two parts: (1) M g has canonical singularities near C, and (2) M g does not have canonical singularities near C. In case (1), we use a generalization of the Reid-Tai criterion that can be applied to singular toric varieties (we review this generalization of Reid-Tai in the appendix), and we obtain that Spec R the argument is very similar to that in [BFV12], and establishes that Spec R Aut(I) (C,I) /Stab C (I) (and hencē J d,g ) also has canonical singularities at (C, I) in this case. Technically, since we are able to focus on one automorphism of (C, I) at a time, the argument is broken into somewhat finer pieces than just described, but this captures the main points.…”

“…In this paper we focus on two main problems concerning the birational geometry of these spaces, namely determining the Kodaira dimension, and determining the birational automorphism group. These problems go back at least to Caporaso's work, and have been investigated recently by Farkas and Verra [FV13] and Bini, Fontanari and the third author [BFV12] in special cases.…”

“…Theorem 7.4 was proved by Bini-Fontanari-Viviani in [BFV12] under the assumption that gcd(d + 1 − g, 2g−2) = 1, which is exactly the numerical condition on d and g that guarantees thatJ d,g has finite quotient singularities. When this happens, one can prove Theorem 7.4 by a direct application of the Reid-Tai criterion (see [BFV12,Thm. 4.8]).…”

Geometric Invariant Theory for Polarized Curves

“…We mention that, if char.k/ D 0, d > 4.2g 2/ and g 4, then Q d;g is known to have canonical singularities (see [BFV12] in the case where gcd.d C1 g; 2g 2/ D 1 and [CMKV2] in the general case). This result has been used in loc.…”