Reducible varieties and valuated matroids

A projective algebraic set X in the projective n-space is irreducible if there does not exist a decomposition of X as a union of two strictly smaller projective algebraic sets. In practice, it is not usually easy to determine whether a projective algebraic set is irreducible or not. The aim of this project is to prove the conjecture which aims to say that the information of a projective algebraic set being reducible is always found in the valuated matroid of some degree d corresponding to the degree-d part of the ideal defining our projective algebraic set. We have an explicit result of this conjecture for four different families of projective algebraic sets, and the next step is to check to see if we can work out a proof in more generality. As this is a project in mathematics, it falls under the remit of the Engineering and Physical Sciences Research Council, and there are no official external partners for this project.