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Examples of Smooth Non‐General Type Surfaces in P 4
Author(s) -
Popescu S
Publication year - 1998
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611598000082
Subject(s) - mathematics , degree (music) , conjecture , genus , type (biology) , pure mathematics , combinatorics , mathematics subject classification , ecology , botany , physics , acoustics , biology
Smooth projective varieties with small invariants have received renewed interest in recent years, primarily due to the fine study of the adjunction mapping. Now, through the effort of several mathematicians, a complete classification of smooth surfaces in P 4 has been worked out up to degree 10, and a partial one is available in degree 11. On the other side, recently Ellingsrud and Peskine have proved Hartshorne's conjecture that there are only finitely many families of smooth surfaces in P 4 , not of general type. It is believed that the degree of the smooth, non‐general type surfaces in P 4 should be less than or equal to 15. The aim of this paper is to provide a series of examples of smooth surfaces in P 4 , not of general type, in degrees varying from 12 up to 14, and to describe their geometry. By using mainly syzygies and liaison techniques, we construct the following families of surfaces: (i) minimal proper elliptic surfaces of degree 12 and sectional genus π = 13; (ii) two types of non‐minimal proper elliptic surfaces of degree 12 and sectional genus π = 14; (iii) non‐minimal K 3 surfaces of degree 13 and sectional genus 16; and iv non‐minimal K 3 surfaces of degree 14 and sectional genus 19. 1991 Mathematics Subject Classification : 14M07, 14J25, 14J26, 14J28, 14C05.