We show that certain qualitative properties of classical solutions to the inviscid Proudman-Johnson equation are preserved as long as these solutions exist. This enables us to give a simple blow-up criterion.

In [S1], we showed that the growth function $P_M(t)$ for an Artin monoid associated with a Coxeter matrix $M$ of finite type is a rational function of the form $1/(1 - t)N_M(t)$, where $N_M(t)$ is a polynomial determined by the Coxeter-Dynkin graph for $M$, and is called the denominator polynomial of type $M$. We formulated three conjectures on the zeros of the denominator polynomial. In the present note, we prove that the same denominator formula holds for an arbitrary Artin monoid, and formulate slightly modified conjectures on the zeros of the denominator polynomials of affine types. The new conjectures are verified for types $\tilde A_2, \cdots , \tilde A_8, \tilde C_2, \cdots ,$ $\tilde C_8, \tilde D_4, \tilde E_7, \tilde E_8, \tilde F_4, \tilde G_2$ among others. In Appendix, we define the elliptic denominator polynomials by formally applying the denominator polynomial formula to the elliptic diagrams for elliptic root systems [S2]. Then, the new conjectures are verified also for elliptic denominator polynomials of types $A_2^{(1,1)}, \cdots , A_7^{(1,1)},D_4^{(1,1)}, E_6^{(1,1)}, E_7^{(1,1)},E_8^{(1,1)}$ and $G_2^{(1,1)}$.

For any $n\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\geq 5$, where any nontrivial nef line bundles are big.