Structure of the F-blowups of simple elliptic singularities (Q2788589)

Let \(X\) be a Noetherian integral scheme, and let \(\mathcal{M}\) be a coherent sheaf on \(X\); a modification \(f:\;Y\rightarrow X\) is called the \textit{blowup of \(X\) at \(\mathcal{M}\)} (denoted \(\mathrm {Bl}_{\mathcal{M}} (X)\)) if, on one hand, \(f\) is a flattening of \(\mathcal{M}\) (this means that the torsion free pullback \(f^{\star} (\mathcal{M})=f^*\mathcal{M}/\mathrm{torsion}\) is locally free) and, on the other hand, that any other flattening of \(\mathcal{M}\) factors through \(f\). The reader will note that, when \(\mathcal{M}\) is an ideal sheaf, \(\mathrm {Bl}_{\mathcal{M}} (X)\) is just the usual blowup with respect to an ideal.NEWLINENEWLINEFrom now on, suppose that \(X\) has prime characteristic \(p\), and that the absolute Frobenius \(F: X\rightarrow X\) is finite; for any integer \(e\geq 0\), one can define the \textit{\(e\)th F-blowup of \(X\)} to be \(\mathrm {Bl}_{F_*^e \mathcal{O}_X} (X):=\mathrm {FB}_e (X)\).NEWLINENEWLINEHereafter, suppose that \((X,x)\) is a simple elliptic surface singularity in prime characteristic \(p\) with exceptional (elliptic) curve \(E\) on the minimal resolution \(\widetilde{X}\). Motivated by [Algebra Number Theory 7, No. 3, 733--763 (2013; Zbl 1303.14018)], the main goal of the paper under review is to understand the structure of \(\mathrm {FB}_e (X);\) it turns out that this structure depends on the intersection number \(-E^2\). Indeed, if \(-E^2\) is not a power of \(p\), then \(\mathrm {FB}_e (X)\cong\widetilde{X}\) (see Theorem 1.1). In this way, assume now that \(-E^2=p^n\) is a power of \(p\), let \(P_0\in E\) be the zero element of the group law, and pick an integer \(e\geq 0\) such that \(p^e\geq\max\{3,p^n\}\). One has to distinguish two cases (see Theorem 1.2).NEWLINENEWLINEOn one hand, if \(E\) is ordinary (this means that, when \(q\) is a power of \(p\), the set of all \(q\)-torsion points on \(E\) has exactly \(q\) different points), then \(\mathrm {FB}_e (X)\) coincides with the blow up of \(\widetilde{X}\) at these points if \(n\geq 1\), and is the blow up at these points ruling out \(P_0\) if \(n=0\). On the other hand, if \(E\) is supersingular (this means that, when \(q\) is a power of \(p\), the set of all \(q\)-torsion points on \(E\) boils down to \(P_0\)), then \(\mathrm {FB}_e (X)\) coincides with the blow up of \(\widetilde{X}\) at an ideal defining a fat point at \(P_0\), where \(P_0\in\widetilde{X}\) is defined in local coordinates \(t,u\) by \((t,u^{p^e})\) if \(n\geq 1\), and is defined by \((t,u^{p^e-1})\) if \(n=1\). It is also worth noting that, in any case, \(\mathrm {FB}_e (X)\cong\widetilde{X}\) provided \(1\leq e<n\).