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Multiplicity Loci

Bundles of differential operators and their application to multiplicity loci will be discussed in this post.

Author

Affiliation

Fei Ye

QCC-CUNY

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1 Introduction

Let $X$ be a smooth projective variety and $D$ an integral ample divisor on $X$. Consider the graded algebra $$R:=\underset{k=0}{\overset{\mathrm{\infty }}{⨁}}{H}^0(X,{\mathcal{O}}_X(kD)).$$ A grade linear series $A$ is a graded subalgebra of a graded algebra $R$. For example, let $V$ be a subvariety of $X$ and $$A_k^{\alpha }:=\{s\in H^0(X,{\mathcal{O}}_X(kD))\mid {\mathrm{mult}}_x(s)\ge k\alpha \text{for all}x\in V\}.$$ Then $$A^{\alpha }(V,X):=\underset{k=0}{\overset{\mathrm{\infty }}{⨂}}{A}_k^{\alpha }$$ is a grade linear series.

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

The volume ${\mathrm{Vol}}_X(A)$ 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 ${\mathrm{Vol}}_X(D)=D^n$. We also have ${\mathrm{Vol}}_X(A^{\alpha }(x,X))\ge D^n-{\alpha }^n$, where $x$ is a point in $X$.

In general, to estimate the volume ${\mathrm{Vol}}_X(A^{\alpha }(V,X))$, we use the short exact sequence and pass the estimation to that of the restricted volume ${\mathrm{Vol}}_{X|V}(D)$ of $D$ along $V$ defined as $${\mathrm{Vol}}_{X|V}(D):=\underset{k\to \mathrm{\infty }}{lim sup}\frac{\dim H^0(X|V,{\mathcal{O}}_X(kD))}{k/d!},$$ where $d=\dim V$ and $$H^0(X|V,{\mathcal{O}}_X(kD)):=\mathrm{Im}(H^0(X,{\mathcal{O}}_X(kD))\to H^0(V,{\mathcal{O}}_V(kD))).$$

By (Ein et al. 2009) Corollary 2.15, the $\underset{x\to \mathrm{\infty }}{lim sup}$ can be replace by $\underset{x\to \mathrm{\infty }}{lim}$ and $D$ maybe replaced by any $\mathbb{Q}$–divisor.

By (Ein et al. 2009) Theorem B, the volume ${\mathrm{Vol}}_{X|V}(A)$ 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.

2 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 $$Z_{\sigma }(E_k):=\{x\in X\mid {\mathrm{mult}}_x(E_k)\ge k\sigma \}$$ which is called a multiplicity locus.

By (Lazarsfeld 2004) Proposition 2.1.20, there exists a $m_0$ such that the base loci $|k{m}_{0}D|$ 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 $$Z_{\sigma }(E_{k_1})=Z_{\sigma }(E_{k_2})\text{for all}{k}_1,k_2\ge k_0.$$

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 $$Z(c)\subset Z(a)\text{for all}c\ge k(a).$$ Otherwise, there will be an infinite chain of subvarieties where the inclusions are strict.

Suppose that $x\notin Z(a)$. We will show that $x\notin Z(c)$ for all sufficiently large $c$. Let $n$ be the minimal positive integer such that $n{E}_a$ is an integral divisor and $\eta =\frac{1}{n}$. Then ${\mathrm{mult}}_x{E}_a\le a\sigma -\eta$. 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 $$\begin{array}{rl}{\mathrm{mult}}_x{F}_c=& \alpha {\mathrm{mult}}_x{E}_a+\beta {\mathrm{mult}}_x{E}_b\\ \le & a\alpha \sigma -\alpha \eta +\beta {\mathrm{mult}}_x{E}_b\end{array}$$

By Bertini’s Theorem, we know that for a general divisor $E_c\in A_c$ the following holds $$\begin{array}{rl}{\mathrm{mult}}_x{E}_c\le & {\mathrm{mult}}_x{F}_c+1\\ \le & c\sigma -b\beta \sigma -\alpha \eta +\beta {\mathrm{mult}}_x{E}_b+1\\ =& c(\sigma -\frac{\eta (1-\frac{\beta b}{c})}{a}+\frac{1+\beta {\mathrm{mult}}_x{E}_b-b\sigma }{c})\end{array}$$ Since $\eta$, $\beta$ and $b$ are bounded and independent of $c$, it follows that ${\mathrm{mult}}_x{F}_c<c\sigma$ for all sufficiently large $c$. Therefore, $x\notin Z(c)$.

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 $${\mathrm{mult}}_{Z}D\le {\mathrm{mult}}_{x}D,$$ where ${\mathrm{mult}}_{Z}D$ 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 $$Z_{\sigma }(A)=Z_{\sigma }(E_k)\text{for}k\gg 1.$$

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.

3 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 ${\mathcal{D}}_{mL}^l\otimes {\mathcal{O}}_X(l\delta L)$ 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 $Z_{\sigma +\epsilon }(A)$, where $A\subset \underset{k=0}{\overset{\mathrm{\infty }}{⨁}}{H}^0(X,{\mathcal{O}}_X(kL))$ is a graded liner series, then $V$ is also an irreducible component of the base locus of the linear series $|I_V^{(k\epsilon )}\otimes {\mathcal{O}}_X(k(1+\delta \sigma )L)|$, where $k$ is a sufficiently larger and sufficiently divisible integer, and $$I_V^{(k\epsilon )}=\{f\mid {\mathrm{mult}}_x(f)\ge k\epsilon \text{for all}x\in V\}$$ is the symbolic power.

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

Because $I_V^{(k\epsilon )}\subset I_V$. It is clear that $$V\subset \mathrm{Bs}(|I_V^{(k\epsilon )}\otimes {\mathcal{O}}_X(k(1+\delta \sigma )L)|).$$

Let $s_k\in H^0(X,{\mathcal{O}}_X(kL))$ be the section whose zeroes is the divisor $E_k$.

Set $${\mathrm{\Sigma }}_{k\sigma -1}=\{x\in X\mid {\mathrm{mult}}_x(E_k)>k\sigma -1\}$$ and denote its ideal by $I_{{\mathrm{\Sigma }}_{k\sigma -1}}$. Because ${\mathcal{D}}_{kL}^{k\sigma }\otimes {\mathcal{O}}_X(k\sigma \delta L)$ is globally generated, the image $I_{{\mathrm{\Sigma }}_{k\sigma -1}}\otimes {\mathcal{O}}_X(k\sigma \delta L))$ of the morphism $${\mathcal{D}}_{kL}^{k\sigma }\otimes {\mathcal{O}}_X(k\sigma \delta L)\to {\mathcal{O}}_X(k\sigma \delta L))$$ is also globally generated. Indeed, $$\begin{array}{rl}H=& H^0(X,I_{{\mathrm{\Sigma }}_{k\sigma -1}}\otimes {\mathcal{O}}_X(k\sigma \delta L)))\\ =& \{D(s_k)\mid D\in H^0(X,{\mathcal{D}}_{kL}^{k\sigma }\otimes {\mathcal{O}}_X(k\sigma \delta L))\}.\end{array}$$

We first show that $V\subset {\mathrm{\Sigma }}_{k\sigma -1}$. Note that if $x\notin {\mathrm{\Sigma }}_{k\sigma -1}$, then ${\mathrm{mult}}_x(E_k)M<k\sigma$. Thus, there exsits a differential operator $D\in H^0(X,{\mathcal{D}}_{kL}^{k\sigma }\otimes {\mathcal{O}}_X(k\sigma \delta L))$ such that $D(s_k)={\mathrm{mult}}_x(s_k)-k\sigma <0$. Therefore, $x\notin V$ and $V\subset {\mathrm{\Sigma }}_{k\sigma -1}$.

Now we show that $$\mathrm{Bs}(|I_V^{(k\epsilon )}\otimes {\mathcal{O}}_X(k(1+\sigma \delta )L|)\subset {\mathrm{\Sigma }}_{k\sigma -1}.$$ Because ${\mathrm{mult}}_V(s_k)\ge k(\sigma +\epsilon )$. For any differential operator $D\in H^0(X,{\mathcal{D}}_{kL}^{k\sigma }\otimes {\mathcal{O}}_X(k\sigma \delta L))$, we have $${\mathrm{mult}}_V(D(s_k))\ge k(\sigma +\epsilon )-k\sigma =k\epsilon .$$ Therefore, $$H\subset H^0(X,I_V^{(k\epsilon )}\otimes {\mathcal{O}}_X(k(1+\sigma \delta )L)).$$ It follows that $$\mathrm{Bs}(|I_V^{(k\epsilon )}\otimes {\mathcal{O}}_X(k(1+\sigma \delta )L)|)\subset \mathrm{Bs}(|H|)={\mathrm{\Sigma }}_{k\sigma -1},$$ where the equality follows from the fact that $I_{{\mathrm{\Sigma }}_{k\sigma -1}}\otimes {\mathcal{O}}_X(k\sigma \delta L))$ is globally generated.

By the construction of ${\mathrm{\Sigma }}_{k\sigma -1}$, we know that ${\mathrm{\Sigma }}_{k\sigma -1}\subset Z_{\sigma }(A)$. If $W\supset V$ is an irreducible component of ${\mathrm{\Sigma }}_{k\sigma -1}$, then ${\mathrm{mult}}_x(s_k)\ge k\sigma$ 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 $Z_{\sigma }(A)=V\cup V^{\prime }$ where $V$ and $V^{\prime }$ have no common irreducible components, we will see that $W\subset V^{\prime }$ and end with a contradiction that $V$ become an irreducible component of $V^{\prime }$.

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

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 $${\epsilon }^c{\mathrm{deg}}_{L}V\le L^n-(L^n-{\alpha }^n{)}^{\frac{c}{n}}\cdot (L^n{)}^{1-\frac{c}{n}},$$ where $V$ and $\epsilon$ are the ones defined in Proposition @ref(prp:multiplicity-loci), and $c=\mathrm{codim}V$.

References

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.” https://arxiv.org/abs/1902.07460.