Normality of Rees algebras of generalized mixed product ideals

Let $K$ be a field and $K[x_1,x_{2}]$ the polynomial ring in two variables over $K$ with each $x_i$ of degree $1$. Let $L$ be the generalized mixed product ideal induced by a monomial ideal $I\subset K[x_1,x_2]$, where the ideals substituting the monomials in $I$ are squarefree Veronese ideals. In this paper, we study the integral closure of $L$, and the normality of $\mathcal{R}(L)$, the Rees algebra of $L$. Furthermore, we give a geometric description of the integral closure of $\mathcal{R}(L)$.