Danielewski–Fieseler surfaces

Embeddings of Danielewski surfaces in affine space

“…In Section 4, we study Danielewski surfaces with a trivial Makar-Limanov invariant, that is, Danielewski surfaces S which admits two nontrivial G a,k -actions with distinct general orbits. We obtain the following criterion which generalizes Theorem 5.4 in [4]. We obtain the following description (see 4.7 below).…”

“…[5]). More precisely, see [4], every such bundle ρ is a principal homogeneous bundle under the action of a line bundle p : L →X. These principal L-bundles are uniquely determined by data consisting of an invertible sheaf L onX and aČech 1-cocycle g with values in the dual L ∨ of L for a suitable covering U ofX.…”

“…These principal L-bundles are uniquely determined by data consisting of an invertible sheaf L onX and aČech 1-cocycle g with values in the dual L ∨ of L for a suitable covering U ofX. In turn, the pair (L, g) is encoded in a combinatorial datum consisting of a rooted tree with weighted edges, which we call a weighted tree (see [4, Example 1.6 and Theorem 3.2] and 2.2 below). Here we use weighted trees in a different way to construct embeddings of Danielewski surfaces into affine spaces.…”

“…In Section 2, we review the construction of Danielewski surfaces as locally trivial bundles over the affine line with an n-fold origin given in [4]. Then we associate to every fine k-weighted tree γ a closed affine subscheme S γ = Spec(B γ ) of A 1 k × Spec(k[ ]), and we prove the following result (Theorem 2.9).…”

“…We construct explicit embeddings of Danielewski surfaces [4] in affine spaces. The equations defining these embeddings are obtained from the 2 × 2 minors of a matrix attached to a weighted rooted tree γ .…”