Nonexistence of exceptional bundles on ${\mathbb{P}}^3$ with maximal possible ranks

We prove that on ${\mathbb{P}}^3$ there is no exceptional bundle with rank $r=2d^2+1$ and degree $d$ for every $\mathrm{\mid }d\mathrm{\mid }\ge 4$. In particular, we find a new obstruction for the existence of exceptional bundles other than $r\mathrm{\mid }(2d^2+1)$. We also show that there is no exceptional bundle with rank $27$ and degree $11$ to exhibit another different obstruction.