Multi-Rees algebras and partial Castelnuovo–Mumford regularity

Let $S=K[x_1,...,x_n]$ be a standard graded polynomial ring with arbitrary field $K$. Let $I_1,...,I_r$ be homogeneous ideals of $S$. The multi-Rees algebra $Rees(I_1,...,I_r)=S[I_1{T}_1,...,I_r{T}_1]$ encodes the information of any product like $I_1^{a_1}...I_r^{a_r}$ and is a standard tool in the study of regularity of $I_1^{a_1}...I_r^{a_r}$. In this short talk, we try to calculate the partial regularity of $Rees(I_1,...,I_r)$.