# Bertini’s Theorem

Application of Multiplicities of Divisors in Families.

Fei Ye https://yfei.page (QCC-CUNY)https://qcc.cuny.edu
2020-01-21

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).

Theorem 1 (Bertini’s Theorem with Multiplicities) Let $$X$$ be a smooth variety, $$L$$ an integral divisor on $$X$$, and $$V\subset H^0(X,\O_X(L))$$ a finite-dimensional subspace. If $$D$$ is a general element in the linear system $$|V|=\PP(V)$$, then $\mult_x(D)\le \mult_x(|V|)+1$ for every $$x\in X$$.

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).

## Multiplicities of Divisors in Families along Fibers

Definition 1 Let $$Z\subset X$$ be an irreducible subvariety and $$F$$ an effective divisor on $$X$$. The multiplicity of $$F$$ along $$Z$$, denoted by $$\mult_Z(F)=\mult_Z(X, F)$$ is the multiplicity $$\mult_x(F)$$ at a general point $$x\in Z$$.

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,

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

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.

We prove part two by contradiction. Suppose for every $$t$$ in an affine open set $$T'\subset T$$, there always exists a $$y\in M_t$$ such that $$\mult_x(F_t)\ne\mult_{(x, t)}(F)$$. Let $$V=\{(x, t)\mid \mult_x(F_t)\ne\mult_{(x, t)}(F), t\in T'\}$$. Then $$V$$ dominates $$T'$$. By part one, we there is a general $$t$$ in $$T'$$ and a general $$y$$ in $$V_t$$ such that $$\mult_yF_t=\mult_{(t,y)}F_t$$. That contradicts with our definition of $$V$$.

## Proof of Bertini’s Theorem

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

Proof (Proof of Bertini’s Theorem with Multiplicities). Let $$\H\subset X\times |V|$$ be the universal divisor. By part one of Lemma 1, we may fix a general point $$0\in |V|$$ so that $\mult_{(x, 0)}(X\times |V|, \H)=\mult_x(X, \H_0)$ for every $$x\in X_t=X$$. Let $$s_0$$ be a section in $$V$$ defining $$\H_0$$. We may choose an affine coordinates centered at $$0$$ so that $$\H$$ is defined by $s=s_0+t_1s_1+\cdots+t_rs_r,$ where $$r=\dim |V|$$. To lighten notations, we will still write $$s_i$$ for the pullback on $$X\times|V|$$. Note now $$\mult_{x, 0}(s)=\mult_x(s_0)$$. We would like to show that there is a basis $$s'_i$$ of $$|V|$$ such that $$\mult_x(s'_i)\ge \mult_x(X, \H_0)-1$$. Define $s'_i=D(s_i)=\frac{\partial}{\partial t_i}s$ for $$i=1,\dots, r$$. Because differentiation lowers the multiplicity by at most one. Then $\mult_{(x,0)}(s_i')\ge \mult_{(x, 0)}(s)-1=m-1.$ Note that $$s_0, s_1',\dots s_r'$$ still form a basis for $$V$$. Take $$D=\H_0$$, Bertini’s theorem with multiplicities is proved.
Corollary 1 Let $$X$$ be a smooth variety, and $$V$$ a linear series on $$X$$. Then a general element $$D\in |V|$$ is smooth away from the base locus of $$|V|$$.
Proof. Let $$x$$ be a point away from the base locus of $$|V|$$. Then $$\mult_x|V|=0$$. Therefore, a general divisor $$D\in |V|$$ has multiplicity $$\mult_xD\le \mult_x|V|+1=1$$. Therefore, $$D$$ is smooth at $$x$$.

## Smoothing Divisors in Families

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).

Proposition 1 ((Ein, Küchle, and Lazarsfeld 1995, Proposition 2.3)) Let $$X$$ and $$T$$ be smooth irreducible varieties with $$T$$ affine, and $$L$$ an integral divisor on $$X$$. Given irreducible subvarieties $Z\subset V\subset X\times T$ such that $$V$$ dominates $$X$$, and a divisor $$F\in |\pr_X^*L|$$ on $$X\times T$$, suppose $l=\mult_Z(F)\qquad\text{and}\qquad k=\mult_V(F).$ Then there exists a divisor $$E\in |\pr_X^*L|$$ on $$X\times T$$ such that $\mult_Z(E)\ge l-k \qquad\text{and}\qquad V\subsetneq\Supp(E).$
Proof (Sketch of Proof). Let $$\sigma\in\Gamma(X\times T, \pr_X^*L)$$ be a section defining $$F$$. The plan is to show that there is differential operator $$D_\alpha=\sum\limits_{|\alpha|\le k} a_\alpha\frac{\partial^{|\alpha|}}{\partial t^{I}}$$ of order $$\le k$$ on $$T$$ such that $$E=D_\alpha(\sigma)$$ has the required properties. Checking over local charts, it’s not to hard to see that $$D_\alpha(\sigma)$$ is still a section in $$\Gamma(X\times T, \pr_X^*L)$$. The property $$\mult_Z(E)\ge l-k$$ is then automatic for any such $$D_\alpha$$. Because $$V$$ dominates $$X$$, the property $$V\subsetneq \Supp(E)$$ can be proved using similar argument as seen in the proof of Bertini’s theorem. Denote by $$\D^k$$ is the sheaf of differential operators of order $$\le k$$ on $$T$$. It’s is vector bundle on $$T$$,see for example, Proposition 2 in my previous post Bundles of Principal Parts. Because $$T$$ is affine. We may choose finitely may differential operators $$D_\alpha\in \Gamma(T, \D^k)$$ that span $$\D^k$$ at every point of $$T$$. Consider the subset $X\times T\supset B=:\{(x, t)\mid D_\alpha(\sigma)(x,t)=0 ~\text{for all}~ \alpha \},$ Then $$B$$ is the base locus of linear system $$|D_\alpha(\sigma)|$$ spanned by $$D_\alpha(\sigma)$$. Then $$\mult_{(x,t)}|D_\alpha(\sigma)|>0$$ for all $$(x, t)\in B$$. In particular, $$\mult_{x, t}F>k$$ for every $$(x, t)\in B$$. Because $$V$$ dominates $$X$$, for a general $$x\in X$$, we have $$\mult_tF_x=\mult_VF=k$$. Therefore, $$V\subsetneq B$$.

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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