Pym S(2,3)

A quantization of a Poisson structure on \(\mathbb{P}^3\) whose degeneracy divisor is a configuration of three planes and a cubic. The variables \(x_1, x_2, x_3\) commute among themselves, while the fourth variable \(x_4\) acts on them through a derivation built from the cyclic data \((c_i, d_i)\). The point scheme is a union of three lines and a rational normal quartic.

Relations

\(x_4 x_1 - x_1 x_4 - x_1^2 - x_1 ((-c_3 - 2) x_2 + c_1 x_3) - d_1 x_2 x_3\)
\(x_4 x_2 - x_2 x_4 - x_2^2 - x_2 ((-c_1 - 2) x_3 + c_2 x_1) - d_2 x_3 x_1\)
\(x_4 x_3 - x_3 x_4 - x_3^2 - x_3 ((-c_2 - 2) x_1 + c_3 x_2) - d_3 x_1 x_2\)
\(x_2 x_3 - x_3 x_2\)
\(x_3 x_1 - x_1 x_3\)
\(x_1 x_2 - x_2 x_1\)

Parameters

\(c_1, c_2, c_3, d_1, d_2, d_3\) — Six scalars; \(b_i = -c_{i-1} - 2\) are determined by the \(c_i\).

Point scheme

three lines and a rational normal quartic (dimension \(1\)).

Normal elements

normal locus dimension \(-1\) (degree 1), \(-1\) (degree 2).

Hochschild cohomology

\(\mathrm{HH}^\bullet_0(A)\) \(= (1, 4, 9, 9)\)
\(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) \(= (1, 3, 6, 8)\)

Kodaira–Spencer

rank \(9\); injective: no; surjective: yes.

Introduced

2015, MR3366864 . One of Pym's quantum deformations of projective 3-space, arising from a quadratic Poisson structure (the type \(S(2,3)\)).

References

MR3366864 , arXiv:2511.08390

Code

The presentation, ready to paste into a computer algebra system:

needsPackage "AssociativeAlgebras"
K = frac(QQ[c1, c2, c3, d1, d2, d3]);
A = K<|x1,x2,x3,x4|>;
I = ideal {
  x4*x1 - x1*x4 - x1^2 - x1*((-c3 - 2)*x2 + c1*x3) - d1*x2*x3,
  x4*x2 - x2*x4 - x2^2 - x2*((-c1 - 2)*x3 + c2*x1) - d2*x3*x1,
  x4*x3 - x3*x4 - x3^2 - x3*((-c2 - 2)*x1 + c3*x2) - d3*x1*x2,
  x2*x3 - x3*x2,
  x3*x1 - x1*x3,
  x1*x2 - x2*x1
};
B = A/I;

PolyRing := FunctionField(Rationals, ["c1", "c2", "c3", "d1", "d2", "d3"]);;
indets := IndeterminatesOfFunctionField(PolyRing);;
c1 := indets[1];;
c2 := indets[2];;
c3 := indets[3];;
d1 := indets[4];;
d2 := indets[5];;
d3 := indets[6];;
kQ := FreeKAlgebra(PolyRing, 4, "x");;
x1 := kQ.x1;; x2 := kQ.x2;; x3 := kQ.x3;; x4 := kQ.x4;;
rels := [
  x4*x1 - x1*x4 - x1^2 - x1*((-c3 - 2)*x2 + c1*x3) - d1*x2*x3,
  x4*x2 - x2*x4 - x2^2 - x2*((-c1 - 2)*x3 + c2*x1) - d2*x3*x1,
  x4*x3 - x3*x4 - x3^2 - x3*((-c2 - 2)*x1 + c3*x2) - d3*x1*x2,
  x2*x3 - x3*x2,
  x3*x1 - x1*x3,
  x1*x2 - x2*x1
];;
A := kQ / rels;;

// untested (Magma not available here)
K<c1, c2, c3, d1, d2, d3> := RationalFunctionField(Rationals(), 6);
F<x1,x2,x3,x4> := FreeAlgebra(K, 4);
rels := [
  x4*x1 - x1*x4 - x1^2 - x1*((-c3 - 2)*x2 + c1*x3) - d1*x2*x3,
  x4*x2 - x2*x4 - x2^2 - x2*((-c1 - 2)*x3 + c2*x1) - d2*x3*x1,
  x4*x3 - x3*x4 - x3^2 - x3*((-c2 - 2)*x1 + c3*x2) - d3*x1*x2,
  x2*x3 - x3*x2,
  x3*x1 - x1*x3,
  x1*x2 - x2*x1
];
A := quo< F | rels >;

4d-AS-regular

Pym S(2,3)

Code