mapToPunctualHilbertScheme -- maps an ideal into a point in a certain punctual Hilbert scheme

This method maps a P-primary ideal Q into a point in a punctual Hilbert scheme. Let $\mathbb{K}$ be a field of characteristic zero and let $P$ be a prime ideal of codimension $c$ in the polynomial ring $R = \mathbb{K}[x_1,\ldots,x_n]$. We write $\mathbb{F}$ for the field of fractions of the integral domain $R/P$. To simplify our notation, perhaps after a linear change of coordinates, we assume that $\{ x_{c+1}, \ldots, x_n \}$ is a maximal independent set of variables modulo $P$.

The main purpose of this method is to reduce the study of arbitrary $P$-primary ideals in $R = \mathbb{K}[x_1,\ldots,x_n]$ to a zero-dimensional setting over the function field $\mathbb{F}$. This reduction is made by parametrizing $P$-primary ideals with the punctual Hilbert scheme $ {\rm Hilb}^m ( \,\mathbb{F}[[y_1,\ldots,y_c]] \,). $ This is a quasiprojective scheme over the function field $\mathbb{F}$. Its classical points are ideals of colength $m$ in the local ring $\mathbb{F}[[y_1,\ldots,y_c]]$.

This method maps a P-primary ideal Q into a unique point in ${\rm Hilb}^m ( \,\mathbb{F}[[y_1,\ldots,y_c]] \,)$ that corresponds with Q. This method can be seen as an implementation of the map $\gamma$ described in Section 2 of the paper Primary ideals and their differential equations.

i1 : R = QQ[x_1, x_2, x_3]

o1 = R

o1 : PolynomialRing

i2 : Q = ideal(x_1^2, x_2^2, x_1-x_2*x_3)

             2   2
o2 = ideal (x , x , - x x  + x )
             1   2     2 3    1

o2 : Ideal of R

i3 : mapToPunctualHilbertScheme Q

                           2
o3 = ideal (hx  - x hx , hx )
              1    3  2    2

                  /    R   \
o3 : Ideal of frac|--------|[hx ..hx ]
                  |(x , x )|   1    2
                  \  2   1 /