Consider the determinantal varieties $X_r$ of $3\times 3$ matrices of rank at most $r$, which are defined by the vanishing of minors of size $r+1$. We illustrate computationally some of the known results about jets.
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Since $X_0$ is a single point, its first jet scheme consists of a single (smooth) point.
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The jets of $X_2$ (the determinantal hypersurface) are known to be irreducible (see Theorem 3.1 in T. Košir, B.A. Sethuraman, Determinantal varieties over truncated polynomial rings [KS05], or Corollary 4.13 in R. Docampo, Arcs on determinantal varieties [Doc13]). Since $X_2$ is a complete intersection and has rational singularities (see Corollary 6.1.5(b) in J. Weyman, Cohomology of vector bundles and syzygies), this also follows from a more general result of M. Mustaţă (Theorem 3.3 in Jet schemes of locally complete intersection canonical singularities).
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As for the case of $2\times 2$ minors, Theorem 5.1 in [KS05], Theorem 5.1 in C. Yuen, Jet schemes of determinantal varieties, and Corollary 4.13 in [Doc13] all count the number of components; the first two references describe the components further. As expected, the first jet scheme of $X_1$ has two components, one of them an affine space.
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The other component is the so-called principal component of the jet scheme, i.e., the Zariski closure of the first jets of the smooth locus of $X_1$. To check this, we first establish that the first jet scheme is reduced (i.e. its ideal is radical), then use the principalComponent method with the option principalComponent(...,Saturate=>...) set to false to speed up computations.
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Finally, as observed in Theorem 18 of S.R. Ghorpade, B. Jonov and B.A. Sethuraman, Hilbert series of certain jet schemes of determinantal varieties the Hilbert series of the principal component of the first jet scheme of $X_1$ is the square of the Hilbert series of $X_1$.
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