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Another Description of Certain Quartic Double Solids
Author(s) -
Kreussler Bernd
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/(sici)1522-2616(200004)212:1<91::aid-mana91>3.0.co;2-c
Subject(s) - quartic function , mathematics , conic section , bijection , quartic surface , hypersurface , quartic plane curve , pure mathematics , mathematical analysis , combinatorics , geometry
The aim of this short note is to give an explicit description of birationally equivalent models of quartic double solids with at least one node. A quartic double solid is a double covering of ℙ 3 branched along a quartic surface. The nodes of the double solid are in bijection with the nodes of the quartic surface B ⊂ ℙ 3 . Using the projection ℙ 3 → ℙ 2 , whose center is a node of B , we obtain a birational map to a conic bundle over ℙ 2 . We shall explicitly describe a birational equivalence between such a conic bundle and a cubic hypersurface in ℙ 4 . We obtain this by projecting the cubic from a smooth point.