Amit Khetan
"Implicitization of rational surfaces with toric varieties"

A parameterized surface can be represented as a projection from a
toric variety generalizing the classical homogeneous and bihomogeneous
parameterizations. We extend to the toric case two methods for
computing the implicit equation of such a rational parameterized
surface. The first approach uses resultant matrices and gives an exact
determinantal formula for the implicit equation if the
parameterization has no base points. In the case the base points are
isolated local complete intersections, we show that the implicit
equation can still be recovered by computing any non-zero maximal
minor of this matrix.

The second method is the toric extension of the method of moving
surfaces, and involves finding linear and quadratic relations
(syzygies) among the input polynomials. When there are no base
points, we show that these can be put together into a square
matrix whose determinant is the implicit equation. Its extension
to the case where there are base points is also explored.


Modeling and Foundations of multi-sided patches

Part 1: Jorg Peters
Modeling with multisided patches

Part 2: Kestutis Karciauskas
Algebraic constructions of rational multisided patches

Abstract of Part 2:

A careful analysis of the algebraic structure of
rational multi-sided patches in the literature
reveals a deep connection between their geometric
properties and base points of the functions
defining them.

The talk shows how the placement of the base points
of the basis functions of multisided patches helps
to create surfaces with desired geometric properties.


Ragni Piene 
"On the singularities of some parameterized surfaces"

The Pluecker formulas for plane complex algebraic curves give relations
between enumerative characters of the curve, such as its degree and class,
and the numbers of its various singularities. Here we shall look at
similar kinds of formulas for algebraic surfaces in projective 3-space. In
particular we shall consider some parameterized surfaces commonly used in
CAGD: Veronese (triangular), Segre (tensor), Del Pezzo, rational scrolls,
monoids.

In the case of real curves, there is a formula due to Klein and Schuh
involving the number of certain real singularities, and for real surfaces
there is a formula due to Viro.