The Geometry of Siegel Modular Varieties

Hulek, Klaus; Sankaran, G. K.

doi:10.2969/aspm/03510089

Cited by 24 publications

(25 citation statements)

References 76 publications

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“…This was first noticed in [GH,Theorem 1.5 and Remark in Section I] (see also [HS,Section III.3], and see Section 1 for another explanation). Therefore it is natural and interesting to consider the following modified problem.…”

Section: Problem ([S1 Question 5])mentioning

confidence: 64%

Kummer structures on a K3 surface: An old question of T. Shioda

Hosono¹,

Lian²,

Oguiso³

et al. 2003

Duke Math. J.

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Section: Problem ([S1 Question 5])mentioning

confidence: 64%

Kummer structures on a K3 surface: An old question of T. Shioda

Hosono¹,

Lian²,

Oguiso³

et al. 2003

Duke Math. J.

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“…It is easily checked that the groups Γ 2 2 (2, 4) and Γ 2 (2, 4) are normal subgroups of Γ 0 (2) and Γ 0 0 (2) respectively (see (6), (7), (9)). Moreover [Γ 0 (2) : Γ 2 2 (2, 4)] = 96 and the quotient group is isomorphic to the semidirect product F 4 2 ⋉ S 3 where S 3 is the symmetric group of order 3.…”

Section: Isomorphisms Of Modular Threefolds the Main Resultmentioning

confidence: 99%

“…In this section we recall the definition and properties of Siegel modular forms and modular varieties in order to fix notations. For a survey article on Siegel modular varieties we refer to [7].…”

Section: Siegel Modular Varietiesmentioning

confidence: 99%

On isomorphisms between Siegel modular threefolds

Perna

2015

Abh. Math. Semin. Univ. Hambg.

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Abstract. The present paper is motivated by Mukai's "Igusa quartic and Steiner surfaces". There he proved that the Satake compactification of the moduli space of principally polarized abelian surfaces with a level two structure has a degree 8 endomorphism. This compactification can be viewed as a Siegel modular threefold. The aim of this paper is to show that this result can be extended to other modular threefolds.The main tools are Siegel modular forms and Satake compactifications of arithmetic quotients of the Siegel upper-half space. Indeed, the construction of the degree 8 endomorphism on suitable modular threefolds is done via an isomorphism of graded rings of modular forms.By studying the action of the Fricke involution one gets a further extension of the previous result to other modular threefolds.The possibility of a similar situation in higher dimensions is discussed at the end of the paper.

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“…The Boundary of the Level Cover. Recall that Pic Q (A 3 ) = QL ⊕ QD, where L is the first Chern class of the Hodge vector bundle E on A 3 , and D is the class of the boundary divisor (See [HS02]). We further recall (see [ACG11]) that Pic Q (M 3 ) = Qλ ⊕ Qδ 0 ⊕ Qδ 1 .…”

Section: Modular Forms and The Level Covers Ofmentioning

confidence: 99%

The locus of plane quartics with a hyperflex

Hu¹

2016

Proc. Amer. Math. Soc.

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Using the results of [DPFSM14], we determine an explicit modular form defining the locus of plane quartics with a hyperflex among all plane quartics. As a result, we provide a direct way to compute the divisor class of the locus of plane quartics with a hyperflex within M 3 , first obtained in [Cuk89]. Moreover, the knowledge of such an explicit modular form also allows us to describe explicitly the boundary of the hyperflex locus in M 3 . As an example we show that the locus of banana curves (two irreducible components intersecting at two nodes) is contained in the closure of the hyperflex locus. We also identify an explicit modular form defining the locus of Clebsch quartics and use it to recompute the class of this divisor, first obtained in [OS11].

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The Geometry of Siegel Modular Varieties

Cited by 24 publications

References 76 publications

Kummer structures on a K3 surface: An old question of T. Shioda

Kummer structures on a K3 surface: An old question of T. Shioda

On isomorphisms between Siegel modular threefolds

The locus of plane quartics with a hyperflex

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