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Radical decomposition conjecture for the colon ideal of the Jacobian ideal

Assume r=m−2r=m-2, and retain the notation for the degenerated generic matrix, the Jacobian ideal JJ, the colon ideal J:IJ:I, the submatrices Mj,NjM_j,N_j, and the determinants δt,γt\delta_t,\gamma_t from the preceding conjecture. Colon-ideal decomposition conjecture. One has

J:I=(⋂t=1r(xm,1,xm,2,xt+1,m−t+1,δt))∩(⋂t=1r−1(xm−1,3,γ2,xt+1,m−t+1,δt))∩…∩(⋂t=11(xm,1,xm,2,xt+1,m−t+1,δt))∩(x3,m−1,γr,x2,m,δ1).\begin{aligned} J:I={}&\left(\bigcap_{t=1}^{r}(x_{m,1},x_{m,2},x_{t+1,m-t+1},\delta_t)\right)\\ &\cap\left(\bigcap_{t=1}^{r-1}(x_{m-1,3},\gamma_{2},x_{t+1,m-t+1},\delta_t)\right)\cap\ldots\\ &\cap\left(\bigcap_{t=1}^{1}(x_{m,1},x_{m,2},x_{t+1,m-t+1},\delta_t)\right)\\ &\cap(x_{3,m-1},\gamma_{r},x_{2,m},\delta_1). \end{aligned}

In particular, J:IJ:I is a radical ideal.

Sources & referencesView supporting material

Primary source

Rainelly Cunha, Zaqueu Ramos and Aron Simis, “Degenerations of the generic square matrix. Polar map and determinantal structure”, arXiv:1610.07681 (2017).

commutative-algebradeterminantal-ideals

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