Geodesic
An existence theorem for exceptional bundles on $\mathrm K3$ surfaces
Izvestiya. Mathematics , Tome 34 (1990) no. 2, pp. 373-388

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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. Bibliography: 10 titles. 
@article{IM2_1990_34_2_a7,
     author = {S. A. Kuleshov},
     title = {An existence theorem for exceptional bundles on $\mathrm K3$ surfaces},
     journal = {Izvestiya. Mathematics },
     pages = {373--388},
     publisher = {mathdoc},
     volume = {34},
     number = {2},
     year = {1990},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1990_34_2_a7/}
}
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S. A. Kuleshov. An existence theorem for exceptional bundles on $\mathrm K3$ surfaces. Izvestiya. Mathematics , Tome 34 (1990) no. 2, pp. 373-388. http://geodesic.mathdoc.fr/item/IM2_1990_34_2_a7/