General components of tangent spaces and Rees algebras associated with derivations

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Let k k be a field of characteristic zero and V V be a reduced affine k k -variety with tangent space T ( V ) T(V) and arc scheme L ∞ ( V ) \mathscr {L}_{\infty }(V) . In this article, we establish a connection between the Zariski closure G 1 ( V ) \mathscr G_1(V) of the nonsingular locus of the tangent space T ( V ) T(V) of V V and the blow-up of V V along the closed subvariety of V V associated with the datum of the ideal J D J_D generated by the coefficients of any nonzero k k -derivation D D on V V . Precisely, we construct a closed immersion from the affine cone of this blow-up to G 1 ( V ) \mathscr G_1(V) . If dim ⁡ ( V ) ≥ 2 \dim (V)\geq 2 , this results in a strict closed embedding, but it is an isomorphism if dim ⁡ ( V ) = 1 \dim (V)=1 . As an application, when dim ⁡ ( V ) = 1 \dim (V)=1 , we derive a characterization, in terms of the derivations on V V , of the nilpotent regular functions in O ( L ∞ ( V ) ) \mathcal O(\mathscr {L}_{\infty }(V)) that are defined in O ( T ( V ) ) \mathcal O(T(V)) . As a by-product, this isomorphism offers an alternative algorithm for computing a presentation of the ideal formed by such nilpotent functions.