Geometry of decomposable directing modules over tame algebras

Abstract

Let $A$ be a tame algebra and $M$ a directing $A$-module (there exists no sequence $M_1\to \text{...}\to {\tau }_{A}X\to *\to X\to \text{...}\to M_2$ of nonzero maps between indecomposable $A$-modules for some indecomposable nonprojective $A$-module $X$ and indecomposable direct summands $M_1,$$M_2$ of $M$). Then the variety $mo{d}_A\left(\dim M\right)$ of Amodules with dimension vector $\dim M$ is a complete intersection. If, in addition, $M$ is a tilting $A$-module then $mo{d}_A\left(\dim M\right)$ is normal.