Conjecture on the orders of Chow groups A1(X) and A2(X) for affine, simplicial toric varieties

Prove that for any affine, simplicial toric variety X over a field k, the Chow group A1(X) has order equal to |δ|, where δ is the determinant of the matrix whose columns are the minimal generators of the unique maximal cone of the fan of X; moreover, establish that the order of the Chow group A2(X) divides |δ|.

Background

The paper determines structural properties of the Grothendieck group G0(X) for affine, simplicial toric varieties. It proves that G0(X) ≅ Z ⊕ F1G0(X) with F1G0(X) finite, and provides complete computations in low dimensions: for surfaces, G0(X) ≅ Z ⊕ Z/|δ|Z, and for 3-folds, F1G0(X) is an extension of A1(X) by A2(X). In particular, Theorem 4.4 shows that for 3-dimensional affine, simplicial toric varieties, |A1(X)| = |δ|, where δ is the determinant of the matrix formed by the minimal generators of the cone.

Motivated by these results and supported by explicit computations (via SageMath) for several 3-fold examples, the authors formulate a general conjecture asserting that for any affine, simplicial toric variety, |A1(X)| equals |δ| and that |A2(X)| divides |δ|. Here δ is defined using the unique maximal cone of the affine fan, whose minimal generators determine the relevant determinant.