In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let O ( x 1 , … , x n ) {\displaystyle {\mathcal {O}}(x_{1},\ldots ,x_{n})} denote the ring of smooth functions in n {\displaystyle n} variables and f {\displaystyle f} a function in the ring. The Jacobian ideal of f {\displaystyle f} is
J f := ⟨ ∂ f ∂ x 1 , … , ∂ f ∂ x n ⟩ . {\displaystyle J_{f}:=\left\langle {\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right\rangle .}
Relation to deformation theory In deformation theory, the deformations of a hypersurface given by a polynomial f {\displaystyle f} is classified by the ring C [ x 1 , … , x n ] ( f ) + J f . {\displaystyle {\frac {\mathbb {C} [x_{1},\ldots ,x_{n}]}{(f)+J_{f}}}.}
This is shown using the Kodaira–Spencer map.
Relation to Hodge theory In Hodge theory, there are objects called real Hodge structures which are the data of a real vector space H R {\displaystyle H_{\mathbb {R} }} and an increasing filtration F ∙ {\displaystyle F^{\bullet }} of H C = H R ⊗ R C {\displaystyle H_{\mathbb {C} }=H_{\mathbb {R} }\otimes _{\mathbb {R} }\mathbb {C} } satisfying a list of compatibility structures. For a smooth projective variety X {\displaystyle X} there is a canonical Hodge structure.
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