Geometric Properties of the Double-Point Divisor

Details Geometric Properties of the Double-Point Divisor Geometric Properties of the Double-Point Divisor

1995-03-17

arxiv.org/abs/alg-geom/9503009v1

Mathematics

Algebraic Geometry

31 pages. AMSTeX with amsppt

Scientific paper

Let $X^n \subset P^N$ be a nonsingular, nondegenerate projective variety of dimension $n$ and codimension $N-n \ge 2$. Let $|C_X|$ be the linear system determined by the double-point divisor obtained by generically projecting $X$ to a hypersurface in $P^{n+1}$. We classify those varieties for which $C_X$ is not ample, or equivalently, does not separate points of $X$. We call such varieties Roth varieties and prove that they exist for all dimensions $n \ge 2$ and give a description of their properties. For example, in many cases Roth varieties are Castelnuovo varieties. Positivity results for the double-point divisor are analogous to positivity results for the ramification divisor which are studied in adjunction theory.

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