Application of Multiplicities of Divisors in Families.

Let \(X\) be a smooth variety, \(L\) an integral divisor on \(X\), and \(|V|\subset |L|\) a linear system. A frequently seen version of Bertini’s theorem claims that a general member of \(|V|\) is smooth if \(V\) has no base points. The original version of Bertini’s theorem is about multiplicities of divisors (see (Kleiman 1998) for a historical review).

Let \(s_0\), \(s_1\), \(\dots\), \(s_r\) is a basis of \(|V|\). Recall that \[ \mult_x(|V|):=\min_{i}\{\mult_x(s_i)\}. \] Equivalently, \(\mult_x(|V|)\) is the minimum of the multiplicities \(\mult_xE\) of all divisors \(E\) in \(|V|\).

Let’s see a baby example first. Consider the linear system \(|V|\) of lines in \(\PP^2\) passing through \(o:=(0,0,1)\), i.e. lines defined by \(ax+by=0\), where \([a, b]\in\PP^1\). It is clear that \(\mult_o|V|=1\) and \(\mult_x|V|=0\) for \(x\ne o\). On the other hand, \(\mult_xL\le 1\) for any point \(x\) and any member \(L\in |V|\). Although it’s trivial in this example, it is worth to mention that \(\mult_x(\H)=\mult_x(\H_t)\) for a general \(t\in\PP^1\), where \(\H\) is the universal divisor in \(\PP^2\times |V|=\PP^2\times\PP^1\) defined by \(ax+by=0\). Indeed, this observation will be a key in the proof of Theorem 1. We will first proof that the above observation holds true in general.

Note that for non-linear system of divisor, Bertini’s theorem may fail, see the example below (Kleiman 1998, Theorem 3.2).

**Lemma 1 (Multiplicities along fibers) **Let \(p: M\to T\) be a morphism of smooth varieties. Assume that \(V\subset M\) is an irreducible subvariety dominating \(T\). Let \(F\subset M\) be an effective divisor. Then,

- for a general point \(t\in T\), and any irreducible component \(W_t\subset V_t\) in the the fiber \(M_t\), \[\begin{equation} \mult_{W_t}(M_t, F_t)=\mult_{V}(M, F); %(eq:mult-along-fiber-a) \end{equation}\]
- for a general \(t\) and every \(y\in M_t\) \[\begin{equation} \mult_y(M, F)=\mult_y(M_t, F_t). %(eq:mult-along-fiber-b) \end{equation}\]

*Proof. * We prove part one first.
Since the conclusion involves only generality, by replacing \(T\) with an affine open subset, we may assume that \(p: M\to T\) is a smooth morphism by the theorem on generic smoothness (see for example (Vakil 2017, Theorem 25.3.3.)). Moreover, we may assume that \(V\) is smooth and dominating \(T\). By further shrinking \(T\) to a smaller affine open subset, we may assume that \(M=X\times T\) and \(p\) is the projection \(X\times T\to T\) (see for example(authors 2020, Lemma 29.32.14)). Therefore, we may choose local coordinates
\[
(x, t)=(x_1,\dots,x_p,t_1,\dots, t_q)
\]
on \(M=X\times T\) such that \(p(x, t)=t\). Let \(\sigma: T\to X\) be a morphism that defines a section \(T\to X\times T\) by \(t\mapsto (t, \sigma(t))\) such that \(\sigma(T)\subset V\). It suffices to show that
\[
\mult_{\sigma(t)}(F_t)=\mult_{(t,\sigma(t))}(F).
\]
Let \(f(x, t)\) be a local equation of \(F\). Substitute \(x\) by \(x=y+\sigma(t)\) and write
\[
g(y, t)=f(y+\sigma(t), t)=\sum c_\alpha(t)y^\alpha,
\]
where \(\alpha\) is a multi-index and \(c_\alpha(t)\) is a power series in \(t\). Shrink \(T\) again if necessary, we may assume that \(m=\mult_V(F)=\mult_{(p(t), t)}(F)\) for any \(t\in T\). Therefore, the terms of weight \(|T|<m\) in \(g(y,t)\) vanishes because
\[
\frac{\partial^{m-1}}{\partial y_\alpha}g(0,t)=\frac{\partial^{m-1}}{\partial x_\alpha}f(p(t),t)=0.
\]
Then it is enough to choose \(t\) so that \(b_\alpha(t)\ne 0\) for some \(I\) with \(|\alpha|=m\). This completes the proof of part one.

Now we are ready the prove Bertini’s theorem with multiplicities.

Using differentiation in parameter directions to lower the multiplicities of divisors in a family is central technique in many studies on Seshadri constants and related areas. For example, the following proposition plays a central role in (Ein, Küchle, and Lazarsfeld 1995).

For a rigorous proof and a more formal discussion, we refer to (Ein, Küchle, and Lazarsfeld 1995, Proposition 2.3) and (Lazarsfeld 2004, Proposition 5.2.13).

authors, The Stacks project. 2020. “The Stacks Project.” https://stacks.math.columbia.edu.

Ein, Lawrence, Oliver Küchle, and Robert Lazarsfeld. 1995. “Local Positivity of Ample Line Bundles.” *J. Differential Geom.* 42 (2): 193–219.

Kleiman, Steven L. 1998. “Bertini and His Two Fundamental Theorems.” In *Studies in the History of Modern Mathematics. III*, 9–37. Palermo: Circolo Matematico di Palermo.

Lazarsfeld, Robert. 2004. *Positivity in Algebraic Geometry. I*. Vol. 48. Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-642-18808-4.

Vakil, Ravi. 2017. “Foundations of Algebraic Geometry.” http://math.stanford.edu/~vakil/216blog/index.html.

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