Bertini's Theorem

Fei Ye

2020-01-13

Revised on 2021-02-22

Categories: Algebraic Geometry Tags: Bertini theorem Differential operator

Introduction

Let \(X\) be a smooth variety defined over an algebraic closed field \(k\) of characteristic \(0\), \(L\) 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).

In the following, I will follow (Lazarsfeld 2004a, Section 5.2B) to present the theorem with a proof, and discuss other possible applications of the techniques and ideas used in the proof.

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.

In general, for algebraic system of divisors, 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 over a algebraically closed field \(k\) of characteristic \(0\). 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 \(M_t\), \[\begin{equation} \mult_{W_t}(M_t, F_t)=\mult_{V}(M, F); \end{equation}\]
for a general \(t\) and every \(y\in M_t\) \[\begin{equation} \mult_y(M, F)=\mult_y(M_t, F_t). \end{equation}\]

Proof. We prove part one first.

By generic smoothness, after 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(The Stacks Project Authors 2020, Lemma 29.32.14)). Therefore, we may choose coordinates \[ (x, t)=(x_1,\dots,x_p,t_1,\dots, t_q) \] on \(M=X\times T\) such that \(p(x, t)=t\). Since \(V\) deminates \(T\), by further shrinking \(T\) if it is neccessary, there exists a section \(\sigma T\to X\times T\) defined 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 \(c_\alpha(t)\ne 0\) 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 \[ \mult_Z(E)\ge l-k \qquad\text{and}\qquad V\not\subset\Supp(E). \]

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 transition functions 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 \(V\not\subset \Supp(E)\) 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 \(x\in X\) and a general \(t\in T\), we have \(\mult_tF_x=\mult_VF=k\). Therefore, \(V\not\subset B\).

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

Differentiate Divisors in Linear Systems

In the proof of Theorem 1, we used the fact that \(\D^k\) is globally generated because \(T\) is affine. In fact, if we assume that the twisted tangent bundle \(T_X(\delta L)\) is nef (see Definition 6.2.3 in (Lazarsfeld 2004b)), where \(T_X\) is the tangent bundle, \(\delta\) is a positive rational number, and \(L\) is an ample line bundle, then we get a similar result for linear systems (see, for example, Proof of Theorem 2.1 in (Ein, Lazarsfeld, and Nakamaye 1996), or Lemma 1.3 in (Nakamaye 2005))

Let \(\pi: Y\to X\) be the blow-up at a point \(\eta\) with the exceptional divisor \(E\). Given a \(\QQ\)–divisor \(D\) and a positive rational number \(\alpha\), we write \[ D(\alpha)=f^*D-\alpha E, \] and \[ m(\eta,D)=\sup_{\alpha\in \QQ_{\geq 0}}\{D(\alpha) ~\text{is}~ \QQ\text{--effective}\}, \] where \(\mathbf{B}(D)=\bigcap\limits_{k} \mathrm{Bs}|kD|\) is the stable base locus of \(D\). Suppose that \(D\) is a big divisor and \(Z\) is a subvariety of the stable base locus \(\mathbf{B}(D)\). The asymptotic vanishing order \(\mathrm{ord}_Z\lVert D\rVert\) is defined by \[ \mathrm{ord}_Z\lVert D\rVert:=\liminf_{k\to\infty}\dfrac{\mathrm{ord}_ZD_k}{k}, \] where \(D_k\) is a general element in \(\lvert kD \rvert\) for any sufficiently large and divisible \(k\). Let \(Z\) be an irreducible subvariety of \(X\). Denote by \(\tilde{Z}\) the birational transform of \(Z\) in \(Y\). We defined \[ \alpha(Z)=\inf_{\beta\in\QQ}\{\tilde{Z}\subseteq \mathbf{B}(D(\beta))\}. \]

Following the proof of Theorem 2.1 in (Ein, Lazarsfeld, and Nakamaye 1996), we present the following result (see Proposition 4.4 in (Lozovanu 2018)) which is a slight generalization of Lemma 1.3 in (Nakamaye 2005).

Proposition 2 Let \(X\) be a smooth projective variety of dimension \(n\), \(Z\subset X\) an irreducible subvariety, and \(L\) an integral ample divisor. Assume that the sheaf \(\D_{kL}^l\otimes\O_X(l\delta L)\) of differential operators of order \(\le l\) is generated by its sections for a rational number \(\delta\), and all sufficiently large integers \(k\) and \(l\) such that \(l\delta\) is a positive integer. Let \(\beta\) be a rational number in the interval \([\alpha(Z), m(\eta, D)]\) such that with \(\ord_{\tilde{Z}}\lVert D(\beta)\rVert> 0\). Then for any \(\alpha\in[\beta, m(\eta, D))\) we have \[\begin{equation} \ord_{\widetilde{Z}}\lVert D(\alpha) \rVert\geq \alpha-\beta. %+\ord_{\widetilde{Z}}\lVert D(\beta) \rVert. \tag{1} \end{equation}\] In particular, if \(T_X\) is nef, then the inequality (1) holds true.

Proof. Let \(\delta\) be any sufficiently small positive rational number. Then \(T_X(\delta L)\) is ample. We may replace \(T_X\) by \(T_X(\delta L)\) as the conclusion for \(T_X\) can be obtained by taking the limit in \(\delta\). We may assume that \(k\) is sufficiently large and sufficiently divisible so that \(l:=k(\alpha-\beta)\), and \(l\delta\) are both integers. Note that a section \(\sigma\in \Gamma(X, \O_X(kL)\otimes \mfm_x^{k\alpha})\) defines a morphism \[ \D^l_{kL}\otimes\O_X(l\delta L)\to \O_X((k+l\delta)L) \otimes \mathcal{I}_{\Sigma_l}, \] where \[\Sigma_l=\{x\in X\mid D(\sigma)\ge l ~\text{for all}~ D ~\text{in}~ \D^l_{kL}\otimes\O_X(l\delta L) \}\] and \(\mathcal{I}_{\Sigma_l}\) is the ideal sheaf. By the assumption that the sheaf \(\D_{kL}^l\otimes\O_X(l\delta L)\) of differential operators of order \(\le l\) is generated by its sections, we have a morphism of global sections \[ \begin{aligned} d^l: H^0(X, \D_{kL}^l\otimes\O_X(l\delta L))&\to H^0(X, \O_X((k+l\delta)L)\otimes \mathcal{I}_{\Sigma_l})\\ D&\mapsto D(\sigma) \end{aligned}. \] We call the sections in the image of the morphism \(d^l\) differential sections of order \(l\). Then \[ \mult_xD(\sigma)\ge k\alpha-k(\alpha-\beta)=k\beta \] for any differential operator \(D\). Therefore, for any \(D\), the differential section \(D(\sigma)\) is also in \(H^0(X, \O_X((k+l\delta)L)\otimes\mfm_x^{k\beta})\) and hence \[ \mult_ZD(\sigma) \ge k\ord_{\widetilde{Z}}\lVert D(\beta) \rVert. \] Assume that \(\sigma\) is a general section and \(x\in X\) is a general point such that \(\mult_Z\sigma=\mult_x\sigma\). It can be checked that there exists a differential operator of order \(l\) such that \(\mult_xD(\sigma)=\mult_x\sigma-l\). By upper semi-continuity, we see that \(\mult_ZD\le \mult_x\sigma-l\). Therefore, \[ \begin{aligned} \ord_{\widetilde{Z}}\lVert D(\alpha) \rVert=&\frac{\mult_x(\sigma)}{k}\\ =&\frac{l+\mult_x(D(\sigma))}{k}\\ \ge & \alpha-\beta+\ord_{\widetilde{Z}}\lVert D(\beta) \rVert \end{aligned} \] The completes the proof for the first part of the assertion.

Since \(T_X\) is nef and hence \(T_X(\delta L)\) is ample for any positive number \(\delta\), by Lemma 2.5 in (Ein, Lazarsfeld, and Nakamaye 1996), the sheaf \(\D_{kL}^l\otimes\O_X(l\delta L)\) of differential operators of order \(\le l\) is generated by its sections. The proof is then completed.

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———. 2004b. Positivity in Algebraic Geometry. II. Vol. 49. 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]. Berlin: Springer-Verlag. https://doi.org/10.1007/978-3-642-18808-4.

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