## Versal Families and the Existence of Stable Sheaves on a Del Pezzo Surface

J. Math. Sci. Univ. Tokyo
Vol. 3 (1996), No. 3, Page 495--532.

Rudakov, Alexei N.
Versal Families and the Existence of Stable Sheaves on a Del Pezzo Surface
Let $r(F), c_{1}(F), c_{2}(F)$ be rank and Chern classes of an algebraic coherent sheaf $F$ on a Del Pezzo surface $X$. We will call a tuple $\bar c(F)=(r(F),c_{1}(F),c_{2}(F))$ the Chern datum for the sheaf $F$. In the paper we write down several necessary conditions on the Chern datum of a non-exceptional stable sheaf on a Del Pezzo surface which generalize the conditions found by Drezet and Le Potier for sheaves on $\PP{2}$ and we use them to define a set $\mx{\tt D}_{X}$. After that we prove that if $\bar c(F) \in \mx{\tt D}_{X}$ and $F$ can be included in a smooth restricted versal family of sheaves on $X$ then there are stable sheaves in the family so $F$ can be deformed into a stable sheaf with the same Chern datum. We provide a way to construct such families and as an application we prove that for any $c \in \mx{\tt D}_{X}$ it exists a stable sheaf $F$ such that $\bar{c}(F)=c$ provided $X$ is $P_{(1)}^{2}$ -- a Del Pezzo surface which arises by blowing up a point in $\PP{2}$.