Eisenbud's conjecture on secant varieties to Veronese reembeddings

Let X be a subvariety in P^N and fix an integer r. The conjecture of
Eisenbud (for curves, known also as Eisenbud-Koh-Stillman conjecture)
tried to predict a uniform presentation of defining equations of r-th
secant variety to d-th Veronese embedding of X, provided d is
sufficiently large. I will explain the statement in details and I will
present several counterexamples to the conjecture, including curves with
very easy singularities, and a projective space P^N, with N at least 6,
and r at least 14. On the other hand, a much weaker version of the
conjecture is true and I will explain this as well. There is still quite a
big space between the known version and the counterexamples to
investigate. The talk is based on a joint work with Adam Ginensky and
Joseph Landsberg.