Positivity of double point divisors

TL;DR

This paper studies the positivity properties of double point divisors arising from projections of smooth projective varieties, establishing conditions under which these divisors are very ample, base point free, or big, thus advancing understanding of their geometric behavior.

Contribution

It proves that the double point divisor from outer projection is very ample except for Roth varieties, and investigates conditions for the inner projection case to be base point free or big.

Findings

Outer projection double point divisor is very ample except for Roth varieties.

Inner projection double point divisor may not be base point free or ample.

Conditions for inner projection divisors to be base point free or big are identified.

Abstract

The non-isomorphic locus of a general projection from an embedded smooth projective variety to a hypersurface moves in a linear system of an effective divisor which we call the double point divisor. David Mumford proved that the double point divisor from outer projection is always base point free, and Bo Ilic proved that it is ample except for a Roth variety. The first aim of this paper is to show that the double point divisor from outer projection is very ample except in the Roth case. This answers a question of Bo Ilic. Unlike the case of outer projection, the double point divisor from inner projection may not be base point free nor ample. However, Atsushi Noma proved that it is semiample except when a variety is neither a Roth variety, a scroll over a curve, nor the second Veronese surface. In this paper, we investigate when the double point divisor from inner projection is base point free or big.