##
**Quartic double solids with ordinary singularities.**
*(English)*
Zbl 1209.14012

A double solid is a double cover of \(\mathbb{P}^3\) branched along a surface of even degree. This paper deals with ordinary double solids, which are those where the ramification surface has at worst ordinary singularities. In local holomorphic coordinates \(u,v,w,t\), the singularities of such a double solid are thus of three types: \(t^2 = uv\) (type \(A\)), \(t^2 = uvw\) (type \(T\)), and \(t^2 = u^2 -vw^2\) (type \(D\)). The authors describe the mixed Hodge structures on the homology groups of the more general class of ADT threefolds \(X\): they are pure with the exception of \(H^3(X)\), which is an extension of a Hodge structure of weight \(3\) by a Hodge structure of type \((1,1)\), with extension data determined by the Abel-Jacobi mapping to the intermediate Jacobian of the natural resolution of singularities of \(X\). They then study in detail the cyclide double solid (ramified along an irreducible quartic surface whose singular locus is a smooth plane conic), and show that the Torelli mapping, sending \(X\) to the polarized mixed Hodge structure on \(H_3(X)\), is six-to-one.

Reviewer: Christian Schnell (Chicago)

### MSC:

14E20 | Coverings in algebraic geometry |

14J30 | \(3\)-folds |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

PDF
BibTeX
XML
Cite

\textit{M. I. Grooten} and \textit{J. H. M. Steenbrink}, Proc. Steklov Inst. Math. 267, 104--112 (2009; Zbl 1209.14012)

### References:

[1] | I. Dolgachev, ”Topics in Classical Algebraic Geometry. Part 1,” http://www.math.lsa.umich.edu/ idolga/topics1.pdf · Zbl 0284.14003 |

[2] | C. H. Clemens, ”Double Solids,” Adv. Math. 47, 107–230 (1983). · Zbl 0509.14045 |

[3] | A. Dold, Lectures on Algebraic Topology (Springer, Berlin, 1995), Class. Math. · Zbl 0872.55001 |

[4] | P. Griffiths and J. Harris, Principles of Algebraic Geometry (J. Wiley & Sons, New York, 1978). |

[5] | J. de Jong and T. de Jong, ”The Virtual Number of D Points. II,” Topology 29, 185–188 (1990). · Zbl 0752.32011 |

[6] | Y. Namikawa and J. H. M. Steenbrink, ”Global Smoothing of Calabi-Yau Threefolds,” Invent. Math. 122, 403–419 (1995). · Zbl 0861.14036 |

[7] | C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge Structures (Springer, Berlin, 2008), Ergeb. Math. Grenzgeb., 3. Folge 52. |

[8] | G. E. Welters, Abel-Jacobi Isogenies for Certain Types of Fano Threefolds (Math. Centrum, Amsterdam, 1981), Math. Centre Tracts 141. · Zbl 0474.14028 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.