Partial Desingularisations of Quotients of Nonsingular Varieties and their Betti Numbers

Kirwan, Frances

doi:10.2307/1971369

Cited by 165 publications

(323 citation statements)

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“…The result follows from a theorem of F. Kirwan ( §7, [17]) if we prove that the normal space at G 3 (2T ) in a point 2T is isomorphic to H 0 (O P 1 (8)) and that, under this isomorphism, the isotropy group of 2T acts on it as PGL 2 .…”

Section: Blow-upmentioning

confidence: 79%

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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Section: Blow-upmentioning

confidence: 79%

A Compactification of ₃ via K3 Surfaces

Artebani¹

2009

Nagoya Mathematical Journal

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“…In this section we shall describe the construction of the partial resolution of singularitiesM //G → M//G (for more details see [26]). …”

Section: Partial Resolution Of Singularitiesmentioning

confidence: 99%

“…Consider the subset [26], L is the disjoint union of finitely many double cosets for the left N T 1 0 action and the right N R action…”

Section: Pairings On the Partial Desingularizationmentioning

confidence: 99%

“…Remark 36 At this point it is useful to recall from [26], Lemma 7.8 that the fibre at x ∈ Z ss R of the restriction π :M ss → M ss of π toM ss is the semistable stratum of the projective space associated to the normal to GZ ss R in M ss at x with respect to the representation of R on the normal space. Since R ⊆ N R ⊂ G and semistability for some reductive group implies semistability for any reductive subgroup, it follows that in such a fibre semistability for R, for N R and for G are all equivalent.…”

Section: Remark 35mentioning

confidence: 99%

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Cohomology pairings on singular quotients in geometric invariant theory

Jeffrey

Kiem

Kirwan³

et al. 2003

Transformation Groups

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“…The key ingredient of this proposition is Kirwan's blow-up of the GIT quotient [4]. Here we only refer to [7], in particular § §1.1, 1.2, 1.8, for the details.…”

Section: Proposition 24 (O'gradymentioning

confidence: 99%