Application of Bundles of Differential Operators
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.
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.
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.
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