--- title: Higher Nash Blowup in Local Algebra url: https://www.emergentmind.com/topics/higher-nash-blowup-local-algebra type: topic --- # Higher Nash Blowup in Local Algebra Higher Nash blowup local algebra is the body of local algebraic structures used to encode Yasuda’s higher Nash blowup, a refinement of the classical Nash blowup in which singular points are replaced by limits of higher-order infinitesimal neighborhoods of smooth points rather than by limits of tangent spaces alone. In practice, this local algebra is expressed through modules of principal parts, higher Jacobian matrices and their maximal-minor ideals, and quotient algebras such as \(T_n(f)=R/((f)+J_n(f))\) or \(M_n(f)=R/J_n(f)\); in the toric setting it is governed by the semigroup algebra \(k[A]\), the torus-point ideal \(J_0=\langle x^{a_1}-1,\dots,x^{a_s}-1\rangle\), and the Gröbner fan of \(J_n=(J_0)^{n+1}\). These descriptions turn the higher Nash construction from a closure in a Hilbert scheme or Grassmannian into an explicit local-algebraic problem [1305.5661][1411.2676][2310.07976]. ## 1. Geometric origin and principal-parts formalism For a variety \(X\) of dimension \(d\), the \(n\)-th infinitesimal neighborhood of a smooth point \(x\in X\) is \[ x^{(n)}:=\operatorname{Spec}(\mathcal O_{X,x}/m_x^{n+1}), \] and its length is \[ N=\binom{d+n}{d}. \] This yields a map from the smooth locus to the Hilbert scheme, \[ \delta_n:X_{\mathrm{sm}}\to \operatorname{Hilb}_N(X),\qquad x\mapsto [x^{(n)}], \] and the \(n\)-th higher Nash blowup is the closure of the graph of \(\delta_n\). An equivalent Grassmannian description uses the diagonal ideal \(I=\ker(R\otimes R\to R)\) and the higher tangent space \[ T_x^nX:=(I_x/I_x^{n+1})^\vee, \] which has dimension \(\binom{d+n}{d}-1\) at a nonsingular point; for \(n=1\), this recovers the classical Nash blowup [1305.5661]. The local algebra underlying this construction is the algebra of principal parts. In the global language of the diagonal, one uses \[ P^n=\mathcal O_{X\times X}/I_\Delta^{n+1}. \] Locally, for a ring \(R\), one writes \[ P_{R/k}^n=(R\otimes R)/I_R^{n+1},\qquad P_{R/k,+}^n=I_R/I_R^{n+1}, \] where \(I_R=\ker(R\otimes R\to R)\). Recent work emphasizes the torsion-free quotient \[ \widetilde{P}_{R,+}^n=P_{R,+}^n/\operatorname{tors}_R(P_{R,+}^n) \] and the canonical higher derivation \[ d_R^n:R\to P_{R/k,+}^n,\qquad x\mapsto (1\otimes x-x\otimes 1)+I_R^{n+1}, \] because freeness or injectivity properties of these modules are what force higher Nash blowups to be trivial only in the smooth case [2001.10491][2511.08792]. | Setting | Local object | Role | |---|---|---| | General variety | \(P_{R/k}^n\), \(P_{R/k,+}^n\) | principal-parts description | | Hypersurface | \(\operatorname{Jac}_n(f)\), \(J_n(f)\) | explicit blowup center | | Hypersurface germ | \(T_n(f)\), \(M_n(f)\) | local higher Nash algebra | | Normal toric variety | \(J_0\), \(J_n\), \(GF(J_n)\) | toric/combinatorial control | The recurrent numerical invariant is the smooth-case rank \(\binom{d+n}{d}\). In the local theory, this number measures the expected size of principal-parts quotients, higher tangent spaces, and higher differential modules; failure to attain the smooth rank is the algebraic signal of singularity. ## 2. Hypersurfaces: higher Jacobian matrices and explicit blowup ideals For a hypersurface \(X=V(F)\subset \mathbb C^s\), the higher Nash blowup admits a concrete matrix description. If \(p=(a_1,\dots,a_s)\in X\), one considers the maximal ideal \[ \mathfrak a_p=(x_1-a_1,\dots,x_s-a_s)\subset \mathbb C[x_1,\dots,x_s] \] and the vector space \(\mathfrak a_p/\mathfrak a_p^{n+1}\), whose basis is indexed by multi-indices \(\alpha\) with \(1\le |\alpha|\le n\). Writing \[ M=\binom{n+s-1}{s-1},\qquad N=\binom{n+s}{s}, \] the higher-order Jacobian matrix \(\operatorname{Jac}_n(F)\) is an \(M\times (N-1)\) matrix whose rows are \[ r_\beta=\left(\frac{\beta!}{\alpha!}\,\partial^{\alpha-\beta}F\right)_{1\le |\alpha|\le n}, \qquad 0\le |\beta|\le n-1, \] with the convention that \(\partial^{\alpha-\beta}F=0\) if some \(\alpha_i<\beta_i\). The identity \[ 0(gF)= \operatorname{Jac}_n(F)\big|_p{}^{\,t}\cdot \big( \partial^\beta g(p) \big)_{0\le |\beta|\le n-1} \] shows that \(\operatorname{Jac}_n(F)\) governs the relations among \(n\)-jets along the hypersurface. For \(n=1\), it is the usual Jacobian matrix [1411.2676]. This matrix controls both geometry and singularity detection. At a smooth point, \[ \operatorname{rank}\operatorname{Jac}_n(F)\big|_p=M, \] and conversely rank \(M\) implies smoothness. Moreover, \[ T_p^nX=\ker \operatorname{Jac}_n(F)\big|_p. \] If \(J_n\subset R\) denotes the ideal generated by the maximal minors of \(\operatorname{Jac}_n(F)\), then the higher Nash blowup of the hypersurface is the blowup of \(J_n\). In parallel, a sheaf-theoretic formulation defines \[ J_n(f):=\operatorname{Fitt}_r(\Omega_f^{(n)}), \] where \(r=\binom{d+n}{n}-1\), and in the affine polynomial case \(J_n(f)\) is generated by the maximal minors of \(\operatorname{Jac}_n(f)\). One paper notes that several versions of the higher Jacobian matrix occur in the literature: in its convention the diagonal entries are \(0\), whereas some older conventions place \(f\) on the diagonal; the two versions agree modulo \(f\) [1411.2676][2310.07976][2412.07254]. The same matrix formalism gives a computational description of the fiber over a singular point. If \(A\) is the set of maximal minors of \(\operatorname{Jac}_n(F)\), one forms \[ \mathcal A=(F,\ u_J-t\,A_J\mid J\in A), \] and the limiting higher tangent spaces are extracted from the elimination ideal \[ V\!\left(\mathcal A \cap \mathbb C[x_1,\dots,x_s,\{u_J\}]\right)\big/\{x_1,\dots,x_s\}. \] This makes the higher Nash fiber a concrete elimination problem rather than a purely abstract closure construction [1411.2676]. ## 3. The local algebra \(T_n(f)\), grading, and derivations For hypersurface germs, the phrase “higher Nash blowup local algebra” usually refers to a quotient by the higher Jacobian ideal. Hussain–Ma–Yau–Zuo define \[ T_n(f)=\mathbb C\{x_1,\dots,x_s\}/(f,J_n(f)), \] and its derivation algebra \[ L_n(f)=\mathrm{Der}(T_n(f)). \] In that formulation, \(T_n(f)\) reduces to the classical Tjurina algebra at \(n=1\). A different notation, used by Le and Yasuda, writes \[ M_n(f)=\mathbb C\{x\}/(f,J_n(f)) \] for the same type of quotient and emphasizes that \(M_1(f)\) is the usual Milnor algebra. This suggests that notation and normalization conventions are not fully uniform across the literature, even though the central local object is the quotient by \((f,J_n(f))\) [2312.04677][2310.07976]. When \(f\) is weighted homogeneous of weight \(w=(w_1,\dots,w_s)\) and weighted degree \(d\), the higher Jacobian ideal inherits the grading. More precisely, the ideal generated by the maximal minors of the higher-order Jacobian matrix is weighted homogeneous for every \(n\). Consequently, \[ T_n(f)=\mathbb C\{x_1,\dots,x_s\}/(f,J_n(f)) \] inherits a grading whenever \(f\) is weighted homogeneous, and if \(f\) defines an isolated hypersurface singularity, then \(T_n(f)\) is Artinian. The associated derivation algebra decomposes into graded pieces \[ L(A)=\bigoplus_k L_k,\qquad L_k=\{D\in L(A)\mid D(A_i)\subseteq A_{i+k}\ \forall i\}. \] The conjecture of Hussain–Ma–Yau–Zuo asserted that under \[ d\ge 2w_1>\cdots>2w_s>0, \] the algebra \(T_n(f)\) admits no nonzero negative weight derivation. That conjecture was proved in dimension two for all \(n\ge 3\), and later a full theorem established that if \(f\in\mathbb C[x_1,\dots,x_s]\) is weighted homogeneous, defines an isolated hypersurface singularity, and satisfies \[ d\ge 2w_1\ge 2w_2\ge \cdots \ge 2w_s>0, \] then \[ T_n(f)=\mathbb C[x_1,\dots,x_s]/\langle f,J_n(f)\rangle \] admits no negative weighted derivations for all \(n\ge 2\). A key ingredient is the lower bound \(\deg_w(g)\ge d\) for every maximal minor \(g\) of \(\operatorname{Jac}_n(f)\), obtained using the containment \(J_n(f)\subseteq J_1(f)^3\) together with the regular-sequence structure of the partial derivatives [2312.04677][2412.07254][2508.14353]. These results give the local algebra a rigidity property parallel to classical statements about Tjurina and Yau algebras. In the weighted homogeneous case, the grading is not an auxiliary decoration but a structural constraint strong enough to exclude degree-lowering derivations. ## 4. Normal toric varieties: semigroup algebra, Gröbner fans, and local combinatorics For a normal toric variety \[ X=\operatorname{Spec}k[A],\qquad A=\sigma\cap \mathbb Z^d, \] with \[ k[A]=k[x^{a_1},\dots,x^{a_s}] \] a monomial subalgebra, the higher Nash blowup admits a particularly explicit local algebraic model. The torus action on \(X\) extends naturally to \(Nash_n(X)\), and the normalized higher Nash blowup is again toric. The distinguished torus-point ideal is \[ J_0=\langle x^{a_1}-1,\dots,x^{a_s}-1\rangle\subset k[A],\qquad J_n=(J_0)^{n+1}, \] which is the algebraic avatar of the \(n\)-th infinitesimal neighborhood of the torus point \(\mathbf 1\). For a weight \(w\in \sigma\), weighted homogenization yields degenerations whose special fibers are the initial ideals \(in_w(J_n)\); thus orbit limits in the higher Nash blowup are encoded by initial ideals of \(J_n\) [1305.5661]. The decisive combinatorial statement is that the fan of the normalization of \(Nash_n(X)\) is exactly the Gröbner fan \(GF(J_n)\). In other words, the normalized higher Nash blowup is determined by which initial ideal \[ in_w(J_n) \] occurs as \(w\) varies in \(\sigma\). This gives a toric analogue of Nobile’s theorem: for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is nonsingular. If \(X\) is singular, the Gröbner fan of \(J_n\) subdivides \(\sigma\) nontrivially, equivalently there exist weights \(w,w'\in \sigma\) with \[ in_w(J_n)\neq in_{w'}(J_n). \] The proof uses reduced Gröbner bases and two local-algebraic lemmas: first, there exists \(h\in J_n\) whose leading term is \((x^{a_i})^n\) for some ray generator \(a_i\); second, for \(p0\). The same phenomenon persists in prime characteristic [1804.11050][2002.07081]. ## 5. Invariance under contact equivalence and Mather–Yau type reconstruction A major development in the local theory is the proof that higher Nash blowup algebras are invariants of contact equivalence. Two hypersurface germs \(f,g\in \mathbb C\{x\}\) are contact equivalent at \(0\) if \[ g=u\cdot \phi(f) \] for an automorphism \(\phi\) of \(\mathbb C\{x\}\) and a unit \(u\in \mathbb C\{x\}^\times\). Under this hypothesis, one has for every \(n\ge 1\) \[ \mathbb C\{x\}/(f,J_n(f))\cong \mathbb C\{x\}/(g,J_n(g)). \] The proof combines two local statements: automorphisms carry \(J_n(f)\) to \(J_n(\phi(f))\), and multiplication by a unit preserves the ideal \((f,J_n(f))\). In a broader analytic/geometric framework, right equivalence implies \[ M_n(f)\cong M_n(g), \] and contact equivalence implies \[ T_n(f)\cong T_n(g) \] for systems of regular elements in analytic or geometric rings [2310.07976][2412.07254]. This invariance becomes a reconstruction theorem in the Mather–Yau style. For \(R=k\{x\}\) with \(\operatorname{char}(k)=0\), or \(R=k[x]\), and hypersurface equations \(f,g\in R\), the following are equivalent: \[ f\sim_c g, \] \[ T_n(f)\cong T_n(g)\ \text{for all } n\ge 2, \] and \[ T_n(f)\cong T_n(g)\ \text{for some } n\ge 2. \] Thus a single higher Nash blowup algebra of order at least \(2\) determines the contact equivalence class, and hence the isomorphism type of the local ring of the hypersurface singularity. The converse direction uses containment estimates for higher Jacobian ideals, Samuel’s theorem to pass from congruence modulo \(mJ(f)^2\) to coordinate equivalence, and Artin approximation in the convergent case [2412.07254]. This changes the status of higher Nash blowup algebras from auxiliary invariants to complete invariants for hypersurface contact classes in the stated settings. The role played classically by the Tjurina algebra is thereby extended to the family of quotients \(T_n(f)\) for \(n\ge 2\). ## 6. Smoothness criteria, curve semigroups, and limits of the method Higher Nash local algebra also yields smoothness criteria beyond the toric category. For a normal hypersurface \(X\), if \[ \operatorname{Nash}_n(X)\cong X, \] then \(X\) is nonsingular. The local mechanism is that triviality of the higher Nash blowup forces the module of principal parts to split with maximal free rank, while for normal hypersurfaces principal parts are torsion-free; this leaves only the regular case. In the same work, quotient varieties \(\operatorname{Spec}(R^G)\) with suitable linear actions are shown to satisfy \[ \operatorname{Nash}_n(\operatorname{Spec}(R^G))\not\cong \operatorname{Spec}(R^G), \] and for \(F\)-pure varieties triviality of the higher Nash blowup implies strong \(F\)-regularity; in particular, if \(\operatorname{Nash}_1(X)\cong X\), then \(X\) is nonsingular [2001.10491]. A different local approach uses the torsion-free quotient of higher principal parts and a condition denoted \(\mathrm{HN}_n\). In this formulation, for a local ring \(R\) of dimension \(d\), one requires algebraically independent elements \(x_1,\dots,x_d\in m_R\) such that \[ P_{A,+}^n \otimes_A R/m_R \longrightarrow \widetilde{P}_{R,+}^{n} \otimes_R R/m_R \] is injective for \(A=k[x_1,\dots,x_d]\). Using graded arc lemmas, associated graded comparisons, and explicit second-order expansions, one obtains characteristic-free smoothness in the graded case and a characteristic-zero proof for second-order higher Nash blowups: if the higher Nash blowup is an isomorphism, then the variety is smooth under those hypotheses [2511.08792]. The local algebra is also rich enough to describe curve cases in semigroup terms. For a toric curve with numerical semigroup \(T=\{0=s_0