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

Abstract

We prove that on ${\mathbb{P}}^3$ there is no exceptional bundle with rank $r=2d^2+1$ and degree $d$ for every $|d|\ge 4$. In particular, we find a new obstruction 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.

arXiv comment20 pages, 0 figures. In the previous version, I found the claim "$f\cdot g$ has degree -2" on page 7 line 12 is false by constructing an explicit counterexample. The original result is not necessarily false. Using the similar techniques in the previous version and some extra work, new results are obtained in this version