# Ruled Varieties: An Introduction to Algebraic Differential Geometry

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*Gerd Fischer; Jens Piontkowski*

A publication of Vieweg+Teubner

The simplest surfaces, aside from planes, are the traces of a line moving in
ambient space or, more precisely, the unions of one-parameter families of
lines. The fact that these lines can be produced using a ruler explains their
name, “ruled surfaces”. The mechanical production of ruled surfaces
is relatively easy, and they can be visualized by means of wire models. These
models are not only of practical use, but also provide artistic
inspiration.

Mathematically, ruled surfaces are the subject of several branches of
geometry, especially differential geometry and algebraic geometry. In classical
geometry, especially differential geometry and algebraic geometry. In classical
geometry, we know that surfaces of vanishing Gaussian curvature have a ruling
that is even developable. Analytically, developable means that the tangent
plane is the same for all points of the ruling line, which is equivalent to
saying that the surface can be covered by pieces of paper. A classical result
from algebraic geometry states that rulings are very rare for complex algebraic
surfaces in three-space: Quadrics have two rulings, smooth cubics contain
precisely 27 lines, and in general, a surface of degree at least four contains
no line at all. There are exceptions, such as cones or tangent surfaces of
curves. It is also well-known that these two kinds of surfaces are the only
developable ruled algebraic surfaces in projective three-space.

The natural generalization of a ruled surface is a ruled variety, i.e., a
variety of arbitrary dimension that is “swept out” by a moving
linear subspace of ambient space. It should be noted that a ruling is not an
intrinsic but an extrinsic property of a variety, which only makes sense
relative to an ambient affine or projective space. This book considers ruled
varieties mainly from the point of view of complex projective algebraic
geometry, where the strongest tools are available. Some local techniques could
be generalized to complex analytic varieties, but in the real analytic or even
differentiable case there is little hope for generalization: The reason being
that rulings, and especially developable rulings, have the tendency to produce
severe singularities.

As in the classical case of surfaces, there is a strong relationship between
the subject of this book, ruled varieties, and differential geometry. For the
purpose of this book, however, the Hermitian Fubini-Study metric and the
related concepts of curvature are not necessary. In order to detect developable
rulings, it suffices to consider a bilinear second fundamental form that is the
differential of the Gauss map. This method does not give curvature as a number,
but rather measures the degree of vanishing of curvature; this point of view
has been used in a fundamental paper of Griffiths and Harris. One of the
purposes of this book is to make parts of this paper more accessible, to give
detailed and more elementary proofs, and to report on recent progress in this
area.

A publication of Vieweg+Teubner. The AMS is exclusive distributor in North America. Vieweg+Teubner Publications are available worldwide from the AMS outside of Germany, Switzerland, Austria, and Japan.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry and differential geometry.