For square 3 by 3 determinantal varieties, the first jet scheme's Hilbert series equals the square of the base variety's Hilbert series, with analogous formulas for m by m minors and for k-th jets.
Sturmfels, Gr¨ obner bases and Stanley decompositions of determinantal rings, Math
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On jet schemes of determinantal varieties
For square 3 by 3 determinantal varieties, the first jet scheme's Hilbert series equals the square of the base variety's Hilbert series, with analogous formulas for m by m minors and for k-th jets.