On the geometry of the moduli space of spin curves

Ludwig, Katharina

doi:10.1090/s1056-3911-09-00505-0

Cited by 20 publications

(36 citation statements)

References 16 publications

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“…The proof is similar to that of the analogous statement for S g and we refer to [Lud,Theorem 3.1] for a detailed outline of the proof and background on quotient singularities. Locally at [X, η, β], the space R g is isomorphic to a neighbourhood of the origin in C 3g−3 τ /Aut(X, η, β).…”

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 77%

“…Now let σ ∈ Aut(X, η, β) act as a quasi-reflection with eigenvalues ζ and 1. As in the proof of [Lud,Proposition 2.15], there exists a node p 1 ∈ C such that the action of σ is given by σ · τ 1 = ζ τ 1 and σ · τ j = τ j for j = 1. When p 1 ∈ N , the induced automorphism σ C acts via t 1 → ζ 2 t 1 and σ C · t j = t j for j = 1.…”

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

“…[HM,Theorem 1]) and for the moduli space S g of spin curves (cf. [Lud,Theorem 4.1]). We start by explicitly describing the locus of non-canonical singularities in R g , which has codimension 2.…”

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

“…The subgroup Aut 0 (X, η, β) is isomorphic to {±1} CC( X) /±1, where CC( X) is the set of connected components of the non-exceptional subcurve X (compare [CCC,Lemma 2.3.2] and [Lud,Proposition 2.7]). Given γ j ∈ {±1} for every connected component X j of X consider the automorphism γ of η = η | X which is multiplication by γ j in every fibre over X j .…”

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

“…as well as with the actions of the automorphism groups on C 3g−3 τ and C 3g−3 t (see also [Lud,p. 5]).…”

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

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The Kodaira dimension of the moduli space of Prym varieties

Farkas¹,

Ludwig²

2010

J. Eur. Math. Soc.

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129

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Abstract. We study the enumerative geometry of the moduli space R g of Prym varieties of dimension g −1. Our main result is that the compactication of R g is of general type as soon as g > 13 and g is different from 15. We achieve this by computing the class of two types of cycles on R g : one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym-Green conjecture on syzygies of Prym-canonical curves. We also perform a detailed study of the singularities of the Prym moduli space, and show that for g ≥ 4, pluricanonical forms extend to any desingularization of the moduli space.Prym varieties provide a correspondence between the moduli spaces of curves and abelian varieties M g and A g−1 , via the Prym map P g : R g → A g−1 from the moduli space R g parameterizing pairs [C, η], where [C] ∈ M g is a smooth curve and η ∈ Pic 0 (C)[2] is a torsion point of order 2. When g ≤ 6 the Prym map is dominant and R g can be used directly to determine the birational type of A g−1 . It is known that R g is rational for g = 2, 3, 4 (see [Dol] and references therein and [Ca] for the case of genus 4) and unirational for g = 5 (cf.[IGS] and [V2]). The situation in genus 6 is strikingly beautiful because P 6 : R 6 → A 5 is equidimensional (precisely dim(R 6 ) = dim(A 5 ) = 15). Donagi and Smith showed that P 6 is generically finite of degree 27 (cf.[DS]) and the monodromy group equals the Weyl group W E 6 describing the incidence correspondence of the 27 lines on a cubic surface (cf. [D1]). There are three different proofs that R 6 is unirational (cf.[D1], [MM], [V1]). Verra has very recently announced a proof of the unirationality of R 7 (see also Theorem 0.8 for a weaker result). The Prym map P g is generically injective for g ≥ 7 (cf. [FS]), although never injective. In this range, we may regard R g as a partial desingularization of the moduli space P g (R g ) ⊂ A g−1 of Prym varieties of dimension g − 1.The scheme R g admits a suitable modular compactification R g , which is isomorphic to (1) the coarse moduli space of the stack R g = M g (BZ 2 ) of Beauville admissible double covers (cf. [B], [ACV]) and (2) the coarse moduli space of the stack of Prym curves (cf. [BCF]). The forgetful map π : R g → M g extends to a finite map π : R g → M g .

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Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 77%

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

“…as well as with the actions of the automorphism groups on C 3g−3 τ and C 3g−3 t (see also [Lud,p. 5]).…”

Section: The Singularities Of the Moduli Space Of Prym Curvesmentioning

confidence: 99%

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The Kodaira dimension of the moduli space of Prym varieties

Farkas¹,

Ludwig²

2010

J. Eur. Math. Soc.

Self Cite

129

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The birational geometry of $$\overline{{\mathcal {R}}}_{g,2}$$ and Prym-canonical divisorial strata

Bud

2024

Sel. Math. New Ser.

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We prove that the moduli space of double covers ramified at two points $${\mathcal {R}}_{g,2}$$ R g , 2 is uniruled for $$3\le g\le 6$$ 3 ≤ g ≤ 6 and of general type for $$g\ge 16$$ g ≥ 16 . Furthermore, we consider Prym-canonical divisorial strata in the moduli space $$\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$$ C n R ¯ g parametrizing n-pointed Prym curves, and we compute their classes in $$\textrm{Pic}_{\mathbb {Q}}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g)$$ Pic Q ( C n R ¯ g ) .

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Theta Characteristics and Their Moduli

Farkas

2012

Milan J. Math.

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Theta characteristics appeared for the first time in the context of characteristic theory of odd and even theta functions in the papers of Göpel [Go] and Rosenhain [Ro] on Jacobi's inversion formula for genus 2. They were initially considered in connection with Riemann's bilinear addition relation between the degree two monomials in theta functions with characteristics. Later, in order to systematize the relations between theta constants, Frobenius [Fr1], [Fr2] developed an algebra of characteristics 1 ; he distinguished between period and theta characteristics. The distinction, which in modern terms amounts to the difference between the Prym moduli space R g and the spin moduli space S g , played a crucial role in elucidating the transformation law for theta functions under a linear transformation of the moduli, and it ultimately led to a correct definition of the action of symplectic group Sp(F 2g2 ) on the set of characteristics. An overwiew of the 19th century theory of theta functions can be found in Krazer's monumental book [Kr]. It is a very analytic treatise in character, with most geometric applications relegated to footnotes.The remarkable book [Cob] by Coble 2 represents a departure from the analytic view towards a more abstract understanding of theta characteristics using configurations in finite geometry. Coble viewed theta characteristics as quadrics in a vector space over F 2 . In this language Frobenius' earlier concepts (syzygetic and azygetic triples, fundamental systems of characteristics) have an elegant translation. With fashions in algebraic geometry drastically changing, the work of Coble was forgotten for many decades 3 .The modern theory of theta characteristic begins with the works of Atiyah [At] and Mumford [Mu]; they showed, in the analytic (respectively algebraic) category, that the parity of a theta characteristic is stable under deformations. In particular, Mumford's functorial view of the subject, opened up the way to extending the study of theta characteristics to singular curves (which was achieved by Harris [H]), to constructing a proper 1 Frobenius' attempts to bring algebra into the theory of theta functions has to be seen in relation to his famous work on group characters. In 1893, when entering the Berlin Academy of Sciences he summarized his aims as follows [Fr3] : In the theory of theta functions it is easy to set up an arbitrarily large number of relations, but the difficulty begins when it comes to finding a way out of this labyrinth of formulas. Many a distinguished researcher, who through tenacious perseverence, has advanced the theory of theta functions in two

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On the geometry of the moduli space of spin curves

Cited by 20 publications

References 16 publications

The Kodaira dimension of the moduli space of Prym varieties

The Kodaira dimension of the moduli space of Prym varieties

The birational geometry of $$\overline{{\mathcal {R}}}_{g,2}$$ and Prym-canonical divisorial strata

Theta Characteristics and Their Moduli

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