Positivity of mixed multiplicities
  This paper studies mixed multiplicities of an arbitrary standard bigraded algebra and mixed multiplicities of two ideals I, J in a local ring (A,m), where I is an m-primary ideal and J an arbitrary ideal. The main results are criteria for their positivity which can be used to compute them effectively. We also show that the range of positive mixed multiplicities of a bigraded algebra is rigid if the algebra satisfies the first chain condition and is connected in codimension one and that this range is always rigid for mixed multiplicities of ideals. These results can be used to study the mu-invariants of analytic hypersurfaces, the degree of rational varieties obtained by blowing-up projective spaces, and the degree of the Stuckrad-Vogel cycles in intersection theory. 
Ngo Viet Trung