# normalCone -- The normal cone of a subscheme

## Synopsis

• Usage:
normalCone I
normalCone(I,f)
• Inputs:
• I, an ideal,
• f, , optional argument, if given it should be a non-zero divisor in the ideal I
• Optional inputs:
• BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
• DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step. Defaults to infinity
• MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
• PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
• Strategy => ..., default value null, Choose a strategy for the saturation step
• Variable => ..., default value w, Choose name for variables in the created ring
• Outputs:
• a ring, the ring $R[It] \otimes{} R/I$ of the normal cone of $I$

## Description

The normal cone of an ideal $I\subset{} R$ is the ring $R/I \oplus{} I/I^2 \oplus \ldots$, also called the associated graded ring of $R$ with respect to $I$. If $S$ is the Rees algebra of $I$, then this ring is isomorphic to $S/IS$, which is how it is computed here.