# Multiplicity Loci

Application of Bundles of Differential Operators

Fei Ye
June 16, 2020

# Introduction

Let $$X$$ be a smooth projective variety and $$D$$ an integral ample divisor on $$X$$. Consider the graded algebra $R:=\bigoplus_{k=0}^\infty H^0(X,\O_X(kD)).$ A grade linear series $$A$$ is a graded subalgebra of a graded algebra $$R$$. For example, let $$V$$ be a subvariety of $$X$$ and $A_k^\alpha:=\{s\in H^0(X,\O_X(kD))\mid \mult_x(s)\ge k\alpha~\text{for all}~ x\in V\}.$ Then $A^\alpha(V, X):=\bigotimes_{k=0}^\infty A_k^\alpha$ is a grade linear series.

The volume of a grade linear series $$A$$ is defined to be $\Vol_X(A)=\limsup_{k\to \infty}\frac{\dim A_k}{k^n/n!}.$ We simply denote $$\Vol_X(D)$$ for $$\Vol_X(R)$$.

The volume $$\Vol_X(A)$$ measures the degree of freedom in choosing a divisor $$E_k$$ in $$A_k$$ for $$k\gg 0$$.

By asymptotic Riemann-Roch, we know that $$\Vol_X(D)=D^n$$. We also have $$\Vol_X(A^\alpha(x, X))\ge D^n-\alpha^n$$, where $$x$$ is a point in $$X$$.

In general, to estimate the volume $$\Vol_X(A^\alpha(V, X))$$, we use the short exact sequence and pass the estimation to that of the restricted volume $$\Vol_{X|V}(D)$$ of $$D$$ along $$V$$ defined as $\Vol_{X|V}(D):=\limsup_{k\to\infty}\frac{\dim H^0(X|V, \O_X(kD))}{k/d!},$ where $$d=\dim V$$ and $H^0(X|V, \O_X(kD)):=\Im(H^0(X, \O_X(kD))\to H^0(V, \O_V(kD))).$

By (Ein et al. 2009) Corollary 2.15, the $$\limsup_{x\to \infty}$$ can be replace by $$\lim_{x\to \infty}$$ and $$D$$ maybe replaced by any $$\QQ$$–divisor.

By (Ein et al. 2009) Theorem B, the volume $$\Vol_{X|V}(A)$$ computes the rate of growth of number of intersection points $$d$$ general divisors in $$A_k$$ which are away from the base locus of $$A_k$$ for $$k\gg 0$$.

In this notes, we will study multiplicity loci and their applications to volume calculations.

# Multiplicity Loci

For each natural number $$k$$, let $$E_k\in A_k$$ be a general divisor. For any rational number $$\sigma>0$$, we define $Z_\sigma(E_k):=\{x\in X\mid \mult_x(E_k)\ge k\sigma\}$ which is called a multiplicity locus.

By (Lazarsfeld 2004) Proposition 2.1.20, there exists a $$m_0$$ such that the base loci $$|km_0D|$$ are all the same for $$k\ge 1$$. However, it is not in general possible to take $$m_0 = 1$$.

Similarly, but with difference, multiplicity loci stabilize for $$k$$ sufficiently large.

Lemma 1 ((Ein, Lazarsfeld, and Nakamaye 1996) Lemma 3.4) For a fixed positive rational number $$\sigma$$, there exists a positive integer $$k_0$$ such that $Z_\sigma(E_{k_1})=Z_\sigma(E_{k_2})\quad \text{for all}\quad k_1, k_2\ge k_0.$

Proof. We will write $$Z(k)$$ for $$Z_\sigma(E_k)$$.

It suffices to show that for any positive integer $$a$$, there is a positive integer $$k(a)$$ such that $Z(c)\subset Z(a) \quad \text{for all}\quad c\ge k(a).$ Otherwise, there will be an infinite chain of subvarieties where the inclusions are strict.

Suppose that $$x\not\in Z(a)$$. We will show that $$x\not\in Z(c)$$ for all sufficiently large $$c$$. Let $$n$$ be the minimal positive integer such that $$nE_a$$ is an integral divisor and $$\eta=\frac1n$$. Then $$\mult_xE_a\le a\sigma-\eta$$. Let $$b$$ be a positive integer coprime with $$a$$. Then for any integer $$c\ge ab$$, there exist nonnegative integers $$\alpha$$ and $$\beta$$ such that $$c=\alpha a+\beta b$$. We may assume that $$\beta\le \alpha$$. Then the divisor $$F_c:= \alpha E_a +\beta E_b\in |A_c|$$ has the multiplicity \begin{aligned} \mult_xF_c=&\alpha\mult_xE_a+\beta\mult_xE_b\\ \le & a\alpha\sigma-\alpha\eta+\beta\mult_xE_b \end{aligned}

By Bertini’s Theorem, we know that for a general divisor $$E_c\in A_c$$ the following holds \begin{aligned} \mult_xE_c\le & \mult_xF_c+1\\ \le & c\sigma - b\beta\sigma-\alpha\eta+\beta\mult_xE_b+1\\ = & c\left(\sigma -\frac{\eta(1-\frac{\beta b}{c})}{a} + \frac{1+\beta\mult_xE_b-b\sigma}{c}\right) \end{aligned} Since $$\eta$$, $$\beta$$ and $$b$$ are bounded and independent of $$c$$, it follows that $$\mult_xF_c<c\sigma$$ for all sufficiently large $$c$$. Therefore, $$x\not\in Z(c)$$.

The above proof is adapted from (Küchle and Steffens 1999) proof of Lemma 2.3.1.

The original proof of the lemma uses upper semicontinuity lemma which is a corollary of (Lipman 1982) Proposition 3.1 (see also (Smirnov 2019) Corollary 3.5).

Lemma 2 (Upper semicontinuity) Let $$f: X\to S$$ be a morphism with equidimensional fibers. Give an divisor $$D$$ and a point $$x\in X$$, for any subvariety $$Z\subset X$$ such that $$x\in X$$, we have $\mult_ZD\le\mult_x D,$ where $$\mult_ZD$$ is defined at the generic point of $$Z$$.

For a graded linear series $$A$$ and any positive rational number $$\sigma$$, we define the multiplicity locus of $$A$$ by $Z_\sigma(A)=Z_\sigma(E_k) \quad \text{for} \quad k\gg 1.$

For dimension reasons, there is an irreducible subvariety $$V$$ shared by two multiplicity loci. More precisely, we have the following “gap” lemma from (Ein, Lazarsfeld, and Nakamaye 1996). A version that works for a family of divisors can be found in (Küchle and Steffens 1999) Lemma 2.3.2.

Lemma 3 ((Ein, Lazarsfeld, and Nakamaye 1996) Lemma 1.5 and 1.6) Let $$X$$ be a smooth irreducible variety of dimension $$n$$ and $$A$$ a graded linear series. For a sequence of $$n+1$$ numbers $0\le\beta_1\le\beta_1\le\cdots\le\beta_{n+1},$ there is $$0\le i\le n$$ such that $$Z_i$$ and $$Z_{i+1}$$ share an irreducible component $$V$$ of codimension $$i$$ and passing through $$x$$.

We will call the irreducible component shared by two multiplicity loci a multiplicity jumping locus.

# Multiplicity Loci vs Base Loci

In previous section, we’ve learned that there are differences between multiplicity loci and base loci. In this section, we will show that a multiplicity locus may be a base locus for another linear series. This result is from (Ein, Lazarsfeld, and Nakamaye 1996) Theorem 3.9, see also (Küchle and Steffens 1999) Proposition 2.4.1.

Proposition 1 ((Ein, Lazarsfeld, and Nakamaye 1996) Theorem 3.9, see also (Küchle and Steffens 1999) Proposition 2.4.1) Let $$X$$ be a smooth projective variety of dimension $$n$$, $$L$$ an integral ample divisor, and $$\delta$$ a rational number. Assume that the sheaf $$\D_{mL}^l\otimes\O_X(l\delta L)$$ of differential operators of order $$\le l$$ is generated by its sections for sufficiently large integers $$m$$ and $$l$$ such that $$l\delta$$ is a positive integer. If $$V$$ is a multiplicity jumping locus of $$Z_\sigma(A)$$ and $$Z_{\sigma+\varepsilon}(A)$$, where $$A\subset \bigoplus\limits_{k=0}^\infty H^0(X, \O_X(kL))$$ is a graded liner series, then $$V$$ is also an irreducible component of the base locus of the linear series $$|I_V^{(k\varepsilon)}\otimes \O_X(k(1+\delta\sigma)L)|$$, where $$k$$ is a sufficiently larger and sufficiently divisible integer, and $I_V^{(k\varepsilon)}=\{f \mid \mult_x(f)\ge k\varepsilon \quad\text{for all}\quad x\in V\}$ is the symbolic power.

Proof. By Lemma 1 and the assumption, we may assume that for all sufficiently large $$k$$, $$Z_\sigma(A)=Z_\sigma(E_k)$$, $$Z_{\sigma+\varepsilon}(A)=Z_{\sigma+\varepsilon}(E_k)$$ and the sheaf $$\D_{kL}^{k\sigma}\otimes\O_X(k\sigma\delta L)$$ is globally generated.

Because $$I_V^{(k\varepsilon)}\subset I_V$$. It is clear that $V\subset \Bs(|I_V^{(k\varepsilon)}\otimes \O_X(k(1+\delta\sigma)L)|).$

Let $$s_k\in H^0(X,\O_X(kL))$$ be the section whose zeroes is the divisor $$E_k$$.

Set $\Sigma_{k\sigma-1}=\{x\in X\mid\mult_x(E_k)>k\sigma-1\}$ and denote its ideal by $$I_{\Sigma_{k\sigma-1}}$$. Because $$\D_{kL}^{k\sigma}\otimes\O_X(k\sigma\delta L)$$ is globally generated, the image $$I_{\Sigma_{k\sigma-1}}\otimes \O_X(k\sigma\delta L))$$ of the morphism $\D_{kL}^{k\sigma}\otimes\O_X(k\sigma\delta L)\to \O_X(k\sigma\delta L))$ is also globally generated. Indeed, \begin{aligned} H= &H^0(X, I_{\Sigma_{k\sigma-1}}\otimes \O_X(k\sigma\delta L)))\\ =&\{D(s_k)\mid D\in H^0(X, \D_{kL}^{k\sigma}\otimes\O_X(k\sigma\delta L))\}. \end{aligned}

We first show that $$V\subset \Sigma_{k\sigma-1}$$. Note that if $$x\not\in \Sigma_{k\sigma-1}$$, then $$\mult_x(E_k)M<k\sigma$$. Thus, there exsits a differential operator $$D\in H^0(X, \D_{kL}^{k\sigma}\otimes\O_X(k\sigma\delta L))$$ such that $$D(s_k)=\mult_x(s_k)-k\sigma<0$$. Therefore, $$x\not\in V$$ and $$V\subset\Sigma_{k\sigma-1}$$.

Now we show that $\Bs(|I_V^{(k\varepsilon)}\otimes \O_X(k(1+\sigma\delta)L|)\subset\Sigma_{k\sigma-1}.$ Because $$\mult_V(s_k)\ge k(\sigma+\varepsilon)$$. For any differential operator $$D\in H^0(X, \D_{kL}^{k\sigma}\otimes\O_X(k\sigma\delta L))$$, we have $\mult_V(D(s_k))\ge k(\sigma+\varepsilon)-k\sigma=k\varepsilon.$ Therefore, $H\subset H^0(X, I_V^{(k\varepsilon)}\otimes \O_X(k(1+\sigma\delta)L)).$ It follows that $\Bs(|I_V^{(k\varepsilon)}\otimes \O_X(k(1+\sigma\delta)L)|)\subset \Bs(|H|)=\Sigma_{k\sigma-1},$ where the equality follows from the fact that $$I_{\Sigma_{k\sigma-1}}\otimes \O_X(k\sigma\delta L))$$ is globally generated.

By the construction of $$\Sigma_{k\sigma-1}$$, we know that $$\Sigma_{k\sigma-1}\subset Z_\sigma(A)$$. If $$W\supset V$$ is an irreducible component of $$\Sigma_{k\sigma-1}$$, then $$\mult_x(s_k)\ge k\sigma$$ for any $$x\in W$$. It follows that $$W\subset Z_\sigma(A)$$. Consequently, $$W=V$$ is also an irreducible component of $$Z_\sigma(A)$$. Otherwise, write $$Z_\sigma(A)=V\cup V'$$ where $$V$$ and $$V'$$ have no common irreducible components, we will see that $$W\subset V'$$ and end with a contradiction that $$V$$ become an irreducible component of $$V'$$.

Let $$U$$ be an irreducible component of $$\Bs(|I_V^{(k\varepsilon)}\otimes \O_X(k(1+\sigma\delta)L)|)$$ that contains $$V$$. Then $$V\subset U\subset W=V$$ which implies that $$V=U$$ is an irreducible component of $$\Bs(|I_V^{(k\varepsilon)}\otimes \O_X(k(1+\sigma\delta)L)|)$$.

As an application, we end this survey with the following result.

Proposition 2 ((Küchle and Steffens 1999) Proposition 2.5.6) Let $$L$$ be an ample divisor such that $$L^n>\alpha^n$$. The $$L$$-degree of $$V$$ satisfies the following inequality $\varepsilon^c\mathrm{deg}_LV\leq L^n-(L^n-\alpha^n)^{\frac{c}{n}}\cdot(L^n)^{1-\frac{c}{n}},$ where $$V$$ and $$\varepsilon$$ are the ones defined in Proposition 1, and $$c=\mathrm{codim} V$$.

Ein, Lawrence, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa. 2009. “Restricted Volumes and Base Loci of Linear Series.” American Journal of Mathematics 131 (3): 607–51. http://www.jstor.org/stable/40263793.

Ein, Lawrence, Robert Lazarsfeld, and Michael Nakamaye. 1996. “Zero-Estimates, Intersection Theory, and a Theorem of Demailly.” In Higher Dimensional Complex Varieties. Proceedings of the International Conference, Trento, Italy, June 15–24, 1994, 183–207. Berlin: Walter de Gruyter.

Küchle, Oliver, and Andreas Steffens. 1999. “Bounds for Seshadri Constants.” In New Trends in Algebraic Geometry (Warwick, 1996), 264:235–54. London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge. https://doi.org/10.1017/CBO9780511721540.009.

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.

Lipman, Joseph. 1982. “Equimultiplicity, Reduction, and Blowing up.” In Commutative Algebra (Fairfax, Va., 1979), 68:111–47. Lecture Notes in Pure and Appl. Math. Dekker, New York.

Smirnov, Ilya. 2019. “On Semicontinuity of Multiplicities in Families.” http://arxiv.org/abs/1902.07460.

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