Compactifications of the period space of Enriques surfaces II

Sterk, Hjm Hans

doi:10.1007/bf02572623

Cited by 18 publications

(27 citation statements)

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“…For similar results and methods see [32] and [30]. We start considering the case of a one parameter family of smooth quartic curves degenerating to a double conic 2T .…”

Section: Final Extensionmentioning

confidence: 99%

A Compactification of ₃ via K3 Surfaces

Artebani¹

2009

Nagoya Mathematical Journal

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Abstract. S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of M3 to the Baily-Borel compactification of a six dimensional ball quotient. IntroductionThe theory of hypergeometric differential equations developed by Deligne and Mostow in the eighties implies that certain moduli spaces of weighted points in P 1 are isomorphic to quotients of complex balls by the action of a discrete group ([6]). More recently several authors have constructed birational maps between moduli spaces of algebraic varieties such as curves of low genus, Del Pezzo surfaces, cubic threefolds, and complex ball quotients (e.g. [1], [10], [20], [21]). These kinds of correspondences are interesting both because they give a new insight in the compactification problem and because they naturally give an interplay between geometry and arithmetic. In this paper we provide an explicit resolution of the birational map between the moduli space of genus three curves and a six dimensional ball quotient constructed by S. Kondō in [18].Let V = |O P 2 (4)| be the space of plane quartics and V 0 be the open subset parametrizing smooth curves. The degree four cyclic cover of the plane branched along a curve in V 0 is a K3 surface X equipped with an order four non-symplectic automorphism σ. This construction defines a holomorphic period map: where Q 0 is the geometric quotient of V 0 by the action of PGL 3 and M is the moduli space of pairs (X, σ), isomorphic to a six dimensional ball quotient. In [18] S. Kondō shows that P 0 gives an isomorphism between Q 0 and the complement of two irreducible divisors D n , D h in M, corresponding to plane quartics with a node and to smooth hyperelliptic genus three curves respectively. A compactification for the moduli space M is given by the Baily-Borel compactification M * , which in this case is the union of M and one point (see [2]). On the other hand, the moduli space Q 0 is known to be birational to the coarse moduli space M 3 of genus three curves. Natural compactifications for this moduli space are the Deligne-Mumford compactification M 3 and the GIT compactification Q, obtained by taking the PGL 3 -categorical quotient of V .In this paper we construct a distinct compactification Q of M 3 by blowing up the orbit of double conics in Q such that the exceptional divisor is a GIT moduli space of hyperelliptic genus three curves. This compactification parametrizes genus three curves with nodes and cusps and the boundary is a rational curve, parametrizing tacnodal curves. The main theorem is the following Theorem. The map P 0 can be extended to a holomorphic surjective map P : Q −→ M * with the following properties: a) it induces an isomorphism between the locus of stable curves and M; b) the exceptional divisor in Q is mapped isomorphically onto the BailyBorel compactification of D h ; c) the rational curve at the bounda...

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“…For similar results and methods see [32] and [30]. We start considering the case of a one parameter family of smooth quartic curves degenerating to a double conic 2T .…”

Section: Final Extensionmentioning

confidence: 99%

A Compactification of ₃ via K3 Surfaces

Artebani¹

2009

Nagoya Mathematical Journal

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“…Section 8 treats the moduli spaces of 'digonal' K3-surfaces and of Enriques surfaces in this spirit. The last case had already been treated by Sterk [24] in his Ph.D. thesis, so here we limit ourself to describing the connection with his work. Section 10 does not involve geometric invariant theory, but concerns an application to singularity theory: we obtain a precise description of the normalization of the union of the smoothing components of a triangle singularity.…”

Section: Introductionmentioning

confidence: 99%

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

Looijenga¹

2003

Duke Math. J.

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Abstract. We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the proj of an algebra of meromorphic automorphic forms. When that complement has a moduli space interpretation, then what we get is often a compactification obtained by means of geometric invariant theory. We illustrate this with several examples: moduli spaces of polarized K3 and Enriques surfaces and the semi-universal deformation of a triangle singularity.We also discuss the question when a type IV arrangement is definable by an automorphic form.

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“…Our results were announced in [11], but their technical nature was one of the reasons that the details got only partially published (in a preprint form in [12] and in H. Sterk's analysis of the moduli space of Enriques surfaces [16]). We believe that the situation has now changed.…”

Section: Introductionmentioning

confidence: 99%