PomeLemma 11.15.3.EBF7EB73EABE For a family $\mathcal E$ of rank-two vector bundles on $X\times T$ whose fibers have a fixed first Chern class, the parameters $t$ for which $\mathcal E_ t$ is $\mu _ P$-stable form a Zariski-open subset of $T$.
Lemma 11.15.3EBF7EB73EABE
Proof. Fix a stable fiber at $t_0$ and put $\nu =c_1(\mathcal E_ t)\cdot P/2$, which is independent of $t$. Choose $k\geq 1$ such that $\mathcal E_{t_0}^*(kP)$ is globally generated and has vanishing higher cohomology. The argument with Serre vanishing, base change and evaluation used in the previous proof gives, over a neighborhood of $t_0$, maps
\[ \mathcal E_ t\longrightarrow \mathcal O_ X(kP)^{\oplus N} \]
that are injective on every fiber. By the double-dual argument for rank-one subsheaves in the proof of proposition 11.14.1, a destabilizing subsheaf of $\mathcal E_ t$ yields a nonzero map from a line bundle $M=\mathcal O_ X(aS+bf)$ with $\deg _ P M\geq \nu $. At least one component of its composite with the injection above is nonzero, and since effective classes have nonnegative coefficients,
\[ a\leq 4k,\qquad b\leq (12e+3)k. \]
Since $\deg _ P M=(8e+3)a+4b$, the inequality $\deg _ PM\geq \nu $ also gives
\[ a\geq \frac{\nu -4(12e+3)k}{8e+3}, \qquad b\geq \frac{\nu -4k(8e+3)}4. \]
Only finitely many integral pairs $(a,b)$ satisfy these four bounds. For each such pair with $\deg _ P M\geq \nu $, semicontinuity makes the set
\[ \{ t:H^0(X,\mathcal E_ t\otimes M^*)\neq 0\} \]
closed, and it does not contain $t_0$. Conversely, a nonzero map from such an $M$ is injective as a map of sheaves and violates stability. Thus the set of parameters with unstable fibers is, on this neighborhood, exactly the finite union of these closed subsets, and removing the union proves the claim. $\square$
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