## Real and complex varieties: comparisons, contrasts and examples

### Institute for Mathematics and its Applications University of Minnesota April 26, 2007

I want to begin by thanking the IMA administrators and staff for running this incredibly excellent year-long program
on Applications of Algebraic Geometry. And in the same context the organizers of the various workshops, tutorials,
and seminars also deserve to be highly commended.

Outline

Examples and properties of plane curves

•  Standard conics: ellipse, parabola, and hyperbola
 Projective closures and the line at infinity
 A family of cubics

• The hyperbola is disconnected because its projective closure intersects the line at infinity twice.

• Some of the real cubics, however, have the property that their projective closures are disconnected.

• More generally, the projective closure of the real curve y² = f(x) has connected components
that correspond to the intervals where f(x) > 0.

 Hyperbolic paraboloid

 Hyperboloid of one sheet

 A cone asymptotic to a hyperboloid

 The hyperboloid and a tangent plane

• The quadric cone has a unique singular point, namely its vertex.

• Each of the smooth quadric ruled surfaces contains two families of straight lines.

• The intersection of the hyperboloid and the tangent plane is a reducible plane conic
-- accordingly, the union of two lines in the tangent plane.

• Every smooth quadric over C is ruled.

• Non-ruled smooth (real) quadrics: ellipsoid, elliptic paraboloid, hyperboloid of two sheets.
• Sketches not currently available
• The intersection of each surface with a tangent plane is a (singular) conic with only one real point.

Tangent surfaces of space curves.

• The tangent surface of the twisted cubic.
 The image of a rectangular coordinate patch
 Another view, with equal length tangent line segments

• The tangent surface of a curve is singular at all points of the curve itself. In the case of the twisted cubic, these are the only singular points.

• Tangent surfaces of other curves.
• No sketches are presently available.

• Tangent lines at two distinct points can intersect non-trivially.
Hence, these surfaces can have other components of their singular loci
that resemble the ordinary double points of a generic projection.

Images under generic projection.

 Cubic ruled surface

 The Steiner surface

• Given a smooth surface in P4 (projective 4-space), generic projection to P³ refers to the process of centrally projecting it it from a generic point of P4. In the case of a smooth surface in P5, we project from a generic line in P5. {Or we can iterate the process of projecting from a point.}

• The image of a smooth surface, under generic projection to P³, is a surface with a purely 1-dimensional singular locus. Most of the singular points are ordinary double points (where two smooth sheets of the surface cross transversally). There are finitely many pinch points and finitely many triple points.

• For a complex projective variety X, a corollary of the Fulton-Hansen connectedness theorem says that pinch points always occur when X is projected to Pn, where n < 2·dim(X) - 1. {The proof of this corollary involves observing that the pre-image of the diagonal is connected if we map XX to PnPn.}

• There are very few examples of generic projections of smooth surfaces to P³ where triple points are not present. Our cubic ruled surface is one of them; it is the generic projection of a smooth surface of degree 3 in P4.