An existence theorem for exceptional bundles on $\mathrm K3$ surfaces

Discrete invariants of exceptional bundles on a $\mathrm K3$ surface $S$ obey the equation $c_1^2-2r(r-c_2+c_1^2/2)=-2$. In this paper it is proved that if the triple $(r,c_1,c_2)\in\mathbf Z\times\operatorname{Pic}(S)\times\mathbf Z$ satisfies this equation, then there exists an exceptional bundle $E$ on $S$ for which $r(E)=r$, $c_1(E)=c_1$ and $c_2(E)=c_2$ (modulo numerical equivalence). In addition, methods of constructing exceptional bundles on a $\mathrm K3$ surface are indicated.
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