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On smooth rational threefolds of P 5 with rational non—special hyperplane section
Author(s) -
Mezzetti Emilia,
Portelli Dario
Publication year - 1999
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/mana.1999.3212070108
Subject(s) - mathematics , hyperplane , rational surface , section (typography) , projection (relational algebra) , base (topology) , pure mathematics , birational geometry , degree (music) , blowing up , combinatorics , mathematical analysis , algorithm , advertising , business , physics , plasma , quantum mechanics , acoustics
It is known that the smooth rational threefolds of P 5 having a rational non—special surface of P 4 as general hyperplane section have degree d = 3, …, 7. We study such threefolds X from the point of view of linear systems of surfaces in P 3 , looking in each case for an explicit description of a birational map from P 3 to X. For d = 3, … 6 we prove that there exists a line L on X such that the projection map of X centered at L is birational; we completely describe the base loci B of the linear systems found in this way and give a description of any such threefold X as a suitable blowing—down of the blowing—up of P 3 along B. If d = 7, i.e., if X is a Palatini scroll, we prove that, conversely, a similar projection never exists.