The deformation-space isomorphism conjecture for Kalck–Karmazyn algebras

Let $E$subset W be the special fiber consisting of a nodal curve on an algebraic surface, let $\mathop{\operatorname{Def}}_{(E \subset W)}$ be the versal deformation space of the pair, and let $\mathop{\operatorname{Def}}^0_{F_E/\mathscr{E}}$ be the locus of deformations of the Kawamata vector bundle $F_E$ whose endomorphism algebras form a flat deformation of the Kalck–Karmazyn algebra $R_{r,a}=\mathop{\operatorname{End}}(F_E)$. Deformation-space isomorphism conjecture. The map

$$\mathop{\operatorname{Def}}_{(E \subset W)}\to \mathop{\operatorname{Def}}^0_{F_E/\mathscr{E}}$$

is an isomorphism. In particular, these deformation spaces have the equal number of irreducible components. The conjecture would identify the deformation space of the surface–curve pair with the relevant deformation locus of the Kawamata bundle and hence explain the correspondence between components arising from $P$-resolutions and components of the algebra deformation space; its status is not established in the supplied text.