Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture (Q2843978)

Let \(X\) be a smooth projective variety in \(\mathbb{P}^n\). Let \(r\) be a positive integer and \(d\) a sufficiently large integer. The main result of this paper says that the \(r\)-th secant variety \(\sigma_r(v_d(X))\) of the \(d\)-th Veronese embedding \(v_d(X)\) of \(X\) is set theoretically equal to the intersection of \(\sigma_r(v_d(\mathbb{P}^n))\) and the linear span of \(v_d(X)\). The authors check that the statement does not hold for a singular variety \(X\). In this paper the motivations for these questions are very well explained. Along the way a number of interesting conjectures are also presented.