Fletcher's codimension conjecture for weighted threefold complete intersections

Let a weighted projective space (WPS) be a projective space with assigned positive weights, and let a weighted complete intersection be a complete intersection in a WPS. A threefold is canonically embedded when its embedding is given by its canonical divisor, and anticanonically embedded when its embedding is given by its anticanonical divisor. A ${\mathbb Q}$-Fano threefold is a three-dimensional ${\mathbb Q}$-Fano variety, and the codimension is the codimension of the complete intersection in the ambient WPS.

Fletcher's codimension conjecture. There are no canonically embedded threefold complete intersections in WPS of codimension $>5$, and there are no anticanonically embedded ${\mathbb Q}$-Fano threefold complete intersections in WPS of codimension $>3$.

These assertions concern the expected bounds on the codimension of canonical and anticanonical weighted complete-intersection threefolds, classes that provide important examples in higher-dimensional birational geometry. The source attributes them to Fletcher; their resolution is not specified in the supplied text.