For cluster-tilted algebras of Dynkin type class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302939&_mathId=si1.gif&_user=111111111&_pii=S0021869316302939&_rdoc=1&_issn=00218693&md5=43e71405044821a87d7b3f67454ede18" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">, we give a geometric description of the stable Cohen–Macaulay category in terms of tagged arcs in the punctured disc. We also describe the action of the syzygy functor in a geometric way. This description allows us to compute the Auslander–Reiten quiver of the stable Cohen–Macaulay category using tagged arcs and geometric moves.