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 \(M_t\), \[\begin{equation} \mult_{W_t}(M_t, F_t)=\mult_{V}(M, F); %(eq:mult-along-fiber-a) \end{equation}\]
  2. 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.

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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