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

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.