Weighted projective lines of tubular type and equivariantization (Q335593)

A \textit{weighted projective line} is the projective scheme \(\mathbb{X}\) given by weighting a set of points \(\{\lambda_1,\dots,\lambda_t\}\subset \mathbb{P}^1_k\) with respective weights (namely, integers larger than \(1\)) \(p=\{p_1,\dots,p_t\}\).NEWLINENEWLINEAssociated with such data is a rank one abelian group, called the \textit{string group}, with respect to which the homogeneous coordinate algebra of \(\mathbb{X}\) is graded. The string group is defined by \(L(p)=\mathbb{Z}e_1\oplus\cdots\oplus\mathbb{Z}e_t/\left<p_1e_1=\cdots=p_te_t\right>\). This group admits a special element which is called the \textit{dualizing element} denoted \(\omega\).NEWLINENEWLINEUnder investigation in this paper is the \textit{tubular} case, namely, when the dualizing element is torsion; in this situation, its order is one of \(2,3,4,6\).NEWLINENEWLINESince the homogeneous coordinate algebra associated with a weighted projective line is graded by the corresponding string group, the dualizing element acts by degree-shifting on the category of graded modules over the homogeneous coordinate algebra, and therefore on the category \(coh(\mathbb{X})\) of coherent sheaves over \(\mathbb{X}\), which is precisely the Auslander-Reiten translation on this category.NEWLINENEWLINEThis paper investigates the subcategories of \(G\)-equivariant objects, where \(G\subset \left<\omega\right>\) is a subgroup of the (finite) group generated by the dualizing element.NEWLINENEWLINEIt turns out that these categories tend to be equivalent to the full categories of choerent sheaves, namely \(coh(\mathbb{X}')\), for \(\mathbb{X}'\) weighted lines of different (tubular) weighting pattern. The main result of this paper is that such equivalences exist for certain weighted projective lines (whenever the base field is algebraically closed and of characteristic not \(2\) or \(3\)).