-- Ex 1a consider forms of degree 2; there is an easy way to determine the number of squares needed to present them as a sum of squares of linear forms(no elimination needed). Analyse (using CoCoA) the cases of two and three variables to find out a general method.
-- Ex 2a how can we check whether a given cube is a sum of two cubes? what about three or four cubes? is it possible to do this using equations?
-- Ex 3a consider binary forms, i.e. homogeneous elements in k[s,t]. How many squares do we need to write the *generic* degree 2 binary form? What about *any* degree two binary form? (To answer just use one conic curve in the plane)
-- Es 1b how many third powers are needed to write the generic cubic form in four variables as a sum of cubes? Is this the expected result?
-- Ex 2b how many linear forms Li are needed to write the *generic* cubic in two variables as a sum of cubes (Li)^3? What happens if we want to write *any* cubic form as a sum of cubes?
-- Es 3b use Terracini's lemma to show that the higher secant varieties of the rational normal curves of P^3 and P^4 have the expected dimension. What about rational normal curves in P^n for n>4?
--Ex 1c compute the ideal of the Segre variety P^1xP^1 embedded in P^3
--Ex 2c compute the ideal of the variety of secant lines of P^1xP^1xP^1
--Ex 3c consider in P^4 two lines and determinane the ideal of their join (use prametrization for the lines, take a general point on each of the line, join the points, eliminate)
--Ex 4c given a line and a conic in P^4 use Terracini's lemma to compute the dimension of their join. Determine the ideal of the join using elimination. Is there any connection with the Segre variety P^xP^1xP^1?