Construction of varieties of low codimension with applications to moduli spaces of varieties of general type

Abstract

We develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m⩾3$, and any given codimension in $P^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non-reduced schemes called ropes, embedded in $P^N$, and by smoothing them. In the range $3⩽m<N/2$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete intersections. We also go beyond a question of Enriques on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. The canonical map of infinitely many of these simple canonical varieties is finite birational but not an embedding. Finally, we show the existence of components of moduli spaces of varieties of general type (in all dimensions $m$, $m⩾3$) that are analogues of the moduli space of curves of genus $g>2$ with respect to the behavior of the canonical map and its deformations. In many cases, the general elements of these components are canonically embedded and their codimension is in the range of Hartshorne's conjecture.