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Let $X$ be an (classical) Enriques surface. Let $|D|$ be a linear system which has no fixed part. $|D|$ is irreducible and $P_a(D)>1$. If the general member of $|D|$ were not reduced, then by Bertini's theorem we would have that the general member of $|D|$ has the form $p^eF$, with $F$ some integral curve on $X$, where $p$ is the characteristic exponent of base field $k$ and $e\ge 1$. (This is in Algebraic Surfaces (L. Badescu), p. 147.)

How can I use Bertini's theorem?

Can you help me?

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Although this is an old question, I think it is useful to note that this follows by the following version of Bertini's theorem, due to Zariski, which I restate in a somewhat more modern fashion:

Theorem [Zar58, Thm. I.6.3]. Let $X$ be a geometrically irreducible normal variety over an algebraically closed field $k$ of characteristic exponent $p \ge 1$. Let $\Lambda$ be a linear system on $X$, of positive dimension $m$, which is free from fixed components. Suppose $\Lambda$ is not composite with a pencil. Then, every general member $D$ of $\Lambda$ is of the form $p^e\Delta$ for some integer $e \ge 0$, where $\Delta$ is a prime divisor.

Since the proof is written in pre-scheme-theoretic language (it would be nice if someone could help me write it up in scheme-theoretic language!), one common thing to do is to reference the following version of Bertini's theorem due to Jouanolou, which I restate to discuss linear systems:

Theorem [Jou83, Thm. 6.3(4)]. Let $X$ be geometrically integral normal scheme of finite type over an infinite field $k$. Let $\Lambda$ be a linear system on $X$, of positive dimension $m$, which is free from fixed components. Suppose $\Lambda$ is not composite with a pencil. Then, every general member $D$ of $\Lambda$ is geometrically irreducible, although may be non-reduced.

Note that Zariski's theorem is a bit stronger than I stated: the number $p^e$ is the degree of inseparability of the field extension of the function field of the generic member of $\Lambda$ over the function field of the projective space $\Lambda$.

However, for your application, we can use Jouanolou's version: a general member of $\lvert D \rvert$ is of the form $mF$, where $F$ is irreducible and $m$ is some integer (Zariski's version of Bertini's theorem says that $m = p^e$), and the rest of the Bădescu's argument shows that $m$ must equal $1$.

Jouanolou does not state this explicitly, but I think [Jou83, Cor. 6.4.2] can be used to recover the explicit description of $p^e$ in Zariski's book.

References

[Jou83] Jean-Pierre Jouanolou. Théorèmes de Bertini et applications. Progr. Math. 42. Boston, MA: Birkhäuser Boston, Inc., 1983. MR: 725671.

[Zar58] Oscar Zariski. Introduction to the problem of minimal models in the theory of algebraic surfaces. Publ. Math. Soc. Japan 4. Tokyo: Math. Soc. Japan, 1958. MR: 0097403.

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