ON SOME NUMERICAL CHARACTERISTICS OF MULTIDIMENSIONAL FANO VARIETIES

On the basis of homological properties of the homogeneous coordinate ring R(V) of an η-dimensional projective Fano variety V over the field C estimates are obtained for the degrees of generators of the algebra R(V) and the ideal I(V). All possible values are found for the dimension and the degree of a variety V of codimension e in P ^ which is not a complete intersection. We give a description of multidimensional Fano varieties of codimension 4 in P ·^ whose linear sections are canonical curves of genus 6. Bibliography: 12 titles. Introduction Let V be an η-dimensional smooth projection algebraic variety over the field C. If the anticanonical sheaf £(—Ky) = (Ω^)* is ample, then V is called a Fano variety, and the maximal natural number r for which -Ky — rD (where D is a divisor) is called the index of V. We show (see 1.1) that for a Fano variety V of index r the graded rings Rs = @^ZQH°(V,msD) are Cohen-Macaulay rings; furthermore, Rs is a Gorenstein ring if and only if r is a multiple of s. Thus the rings Ra have good homological properties, and the goal of this paper is to use these properties of rings to obtain information on numerical characteristics of V and to classify Fano varieties of small codimension in projective spaces. The homology techniques of commutative algebra used here allows us to give a simpler proof and a strengthening of the result of Arbarello and Sernesi on the degrees of generators of the ideal I(V) (see [4]); we also obtain a much less cumbersome proof of Fujita's result [8] to the effect that each Fano variety of degree 5 and codimension 3 in projective space is a complete intersection in the Grassmann variety G(2,5) and give a generalization of this result (see 2.7). In §1 we consider the problem of estimating the degrees of generators of the rings Rs regarded as algebras over C and the degrees of generators of the ideal I(V) for projectively normal V. In §2 we study Fano varieties of codimension 3 in projective space and determine their numerical characteristic; §3 is devoted to classification of ndimensional Fano varieties of codimension 4 in projective space, whose linear sections are canonical curves of genus 6. 1980 Mathematics Subject Classification (1985 Revision). Primary 14J40, 14M07; Secondary 13H10, 14C20. ©1986 American Mathematical Society 0025-5726/86 $1.00 + $.25 per page