# Bertini theorem for big divisors and klt pairs

Let $X$ be a smooth projective variety and let $D$ be a big $\mathbb Q$-divisor on $X$. Assume that for $m$ large $|mD|$ has no fixed components. Is there a $\mathbb Q$-divisor $D'\equiv D$ so that $(X,D)$ is klt?

For $D$ ample, this follows from the usual Bertini theorem, as we can take $D'=\frac1m H$ for $H\in |mD|$ generic, for $m$ large. Does it make sense for a similar `Bertini-type result' in the above situation?

## 2 Answers

If you want an example where $|mD|$ has no fixed codim-1 components, you can do the following. Let $X \dashrightarrow X^+$ be an Atiyah flop of a curve $C$, and let $H \subset X$ be a general member of a very ample linear system with $H \cdot C \geq 2$. The strict transform $\tilde{H} \subset X^+$ is still big. Let $Y$ be the resolution of the flop with exceptional divisor $E$, and $\bar{H}$ the strict transform of $H$ on $Y$.

On $Y$, we have $K_Y = g^* K_{X^+} +E$, and the strict transform is $\bar{H} = f^*H$ (since $H$ is general, it doesn't contain $C$). It's easy to check $f^*H+aE = g^*\tilde{H}$, where $a = H \cdot C$. Then $K_Y + \bar{H} = (g^*K_{X^+} + E) + f^\ast H = (g^* K_{X^+} + E) + (g^*\tilde{H} - aE)$ i.e. $K_Y + \bar{H} = g^*(K_{X^+} + \tilde{H}) + (1-a)E$. Since $a \geq 2$, the pair $(X^+,\tilde{H})$ is not numerically klt (or even numerically lc, if you take $a>2$). The problem is that $|\tilde{H}|$ is too singular along the flopped curve $C^+$; $\tilde{H}$ is already general in its linear series, so we can't get the multiplicity along $C^+$ to be any smaller, and passing to a multiple doesn't help. (Hopefully I got all my signs right!)

• On the other hand, if you mean that given any $V \subset X$ of arbitrary codimension, there is some $m = m(V)$ so that $|mD|$ does not have $V$ in its base locus, then $D$ has to be nef and everything is OK. – user47305 Mar 3 '15 at 1:54

If $X$ is $\mathbb P ^2$ blown up at a point $O$ with exceptional divisor $E$ and $D=H+E$ where $H$ is the pull-back of the hyperplane class, then any effective divisor $D'\equiv D$ will vanish along $E$ with multiplicity $\geq 1$ (i.e. $D'\geq E$) and hence $\mathcal J(D')\subset \mathcal J(E)=\mathcal O _X(-E)$. In other words, the non-klt locus contains $E$. On the other hand, if $D$ is nef and big, then using Wilson's theorem you can show that $\mathcal J (||D||)=\mathcal O _X$ which implies that $(X,D')$ is klt for some effective $\mathbb Q$-divisor $D'\sim _{\mathbb Q}D$ (this is surely proved somewhere in Lazarsfeld positivity books).

• The Wilson's theorem bit is also essentially Prop. 2.61(3) in Koll\'ar-Mori (and indeed surely in Lazarsfeld, which I don't have handy). – user47305 Mar 3 '15 at 1:10
• Wilson's theorem is Theorem 2.3.9 in Lazarsfeld's book "Positivity in algebraic geometry I". The corollary Hacon is talking about is Proposition 11.2.18 in "Positivity in algebraic geometry II". – diverietti Mar 3 '15 at 9:53
• @ diverietti: Thanks for finding the ref. @ Mark: Thanks for giving the correct answer (I had not noticed the "no fixed components" hypothesis). – Hacon Mar 3 '15 at 16:46