Guan's conjecture on equations of secant varieties of Chow varieties

Let XPWX\in \mathbb{P}W^* be an algebraic variety, let σr(X)\sigma_r(X) denote its rr-th secant variety, and let Iδ(Y)I_\delta(Y) denote the degree-δ\delta component of the ideal of a variety YY. For δ=kr+l\delta=kr+l with 0l<r0\leq l<r, choose δ\vec{\delta} so that

δ1==δl=k+1,δl+1==δr=k.\delta_1=\cdots=\delta_l=k+1,\qquad \delta_{l+1}=\cdots=\delta_r=k.

Let Fδ1,δ2,,δrF_{\delta_1,\delta_2,\ldots,\delta_r} and Aδ,iA_{\vec{\delta},i} be the maps and subspaces defined in the paper. Guan's conjecture.

Iδ(σr(X))=Fδ1,δ2,,δr1(Aδ,1++Aδ,r).I_\delta(\sigma_r(X))=F_{\delta_1,\delta_2,\ldots,\delta_r}^{-1}(A_{\vec{\delta},1}+\cdots+A_{\vec{\delta},r}).

This conjecture proposes a uniform description of the degree-δ\delta equations of secant varieties using the maps arising from the paper's prolongation and polarization constructions. The supplied text gives no resolution, so the conjecture remains open.

Sources & referencesView supporting material

Primary source

Yonghui Guan, “Equations for secant varieties of Chow varieties”, arXiv:1602.04275 (2016).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.