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

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