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Seshadri Constants and Restricted Volumes

The purpose of this post is to show an application of the differentiation technique on multiplicity loci to Seshadri constants.

Author

Affiliation

Fei Ye

QCC-CUNY

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The purpose of this post is to show an application of differentiation technique discussed in the post Multiplicity Loci to Seshadri constants. We mainly follow the paper (Cascini and Nakamaye 2014).

1 Base loci

Let $X$ be a smooth projective variety of dimension $n$ and $D$ be an effective $\mathbb{Q}$-divisor on $X$. We denote by $\mathbf{B}(D):=\bigcap _k\mathrm{Bs}|kD|$ the stable base locus of $D$, that is the intersection of the base loci of the linear systems $|kD|$ for all natural numbers $k$.

The augmented base locus $B_{+}(D)$ of $D$, is defined as $${\mathbf{B}}_{+}(D):=\bigcap _A\mathbf{B}(D-A)$$ where $A$ is an ample divisor such that $D-A$ is a $\mathbb{Q}$-divisor.

Remark. If $C$ is a Seshadri exceptional curve for an ample divisor $D$ at $\eta$, then $C\subset {\mathbf{B}}_{+}(D)$.

Suppose that $D$ is a big divisor and $Z$ is a subvariety of $\mathbf{B}(D)$. The asymptotic vanishing order ${\mathrm{ord}}_Z\Vert D\Vert$ is defined by $${\mathrm{ord}}_Z\Vert D\Vert :=\underset{k\to \mathrm{\infty }}{lim inf}{\displaystyle \frac{{\mathrm{ord}}_Z{D}_k}{k}},$$ where $D_k$ is a general element in $|kD|$ for any sufficiently large and divisible $k$.

For the subvariety $Z\subset X$, we define the restricted volume of $D$ along $Z$ to be $$\mathrm{Vol}(X|Z,D)=\underset{k}{lim sup}{\displaystyle \frac{h^0(X|Z,kD)}{k^n/n!}},$$ where $$H^0(X|Z,kD):=\mathrm{Im}(H^0(X,kD)\to H^0(Z,kD{|}_Z))$$ and $h^0(X|Z,kD)=\dim H^0(X|Z,kD)$.

Restricted volumes have the following properties.

Theorem 1 Let $X$ be a smooth projective variety and $Z$ be a subscheme.

1. $\mathrm{Vol}(X|Z,D)>0$ if $Z$ is not contained in ${\mathbf{B}}_{+}(D)$.
2. $\mathrm{Vol}(X|Z,D)$ is numerical invariant.
3. for $k$ sufficiently large and divisible, and for any general elements $D_{k,1},\dots ,D_{k,d}$ in $|kD|$, we have $$\mathrm{Vol}(X|Z,D)={\displaystyle \frac{\mathrm{\#}(Z\cap D_{k,1}\cap \cdots \cap D_{k,d}\setminus \mathbf{B}(D))}{k^d}},$$ where $d=\dim Z$.
4. Let $f:X^{\prime }\to X$ be a proper and birational morphism and $Z^{\prime }$ be an irreducible subvariety on $X^{\prime }$ such that $f(Z^{\prime })=Z$. Then $$\mathrm{Vol}(X^{\prime }|Z^{\prime },f^{\ast }D)=\mathrm{Vol}(X|Z,D).$$

Lemma 1 Let $f:Y\to X$ be the blow-up at a point $\eta$ with the exceptional divisor $E$. Given a $\mathbb{Q}$-divisor $D$ and a positive rational number $\alpha$, we have $$\mathrm{Vol}(Y,D_{\alpha })=\mathrm{Vol}(X,D)-n{\int }_0^{\alpha }\mathrm{Vol}(Y|E,D(\gamma ))\mathrm{d}\gamma ,$$ where $D(\gamma )=f^{\ast }D-\gamma E$.

Let $X$ be a smooth projective variety, $\eta \in X$ be a very general point, and $D$ be an effective integral divisor on $X$. Let $\pi :Y\to X$ be the blow-up at the point $\eta$ with the exceptional divisor $E$. For a subvariety $Z\subset X$, we denote by $\tilde{Z}\subset Y$ the birational transform of $Z$. Given a $\mathbb{Q}$-divisor $D$ and a positive rational number $\alpha$, we set $$D(\alpha )=f^{\ast }D-\alpha E,$$ $$\alpha (Z)=\underset{\beta \in \mathbb{Q}}{inf}\{\tilde{Z}\subseteq \mathbf{B}(D(\beta ))\},$$ and $$m(\eta ,D)=\underset{\alpha \in {\mathbb{Q}}_{\ge 0}}{sup}\{D(\alpha )\text{is }\mathbb{Q}\text{--effective}\}.$$

When estimating the restricted volume $\mathrm{Vol}(Y|E,D(\gamma ))$, a central technique is differentiation.

Lemma 2 ((Nakamaye 2005, Lemma 1.3)) 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 ${\mathcal{D}}_{kL}^l\otimes {\mathcal{O}}_X(l\delta L)$ of differential operators of order $\le l$ is generated by its sections for a rational number $\delta$, and 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 ${\mathrm{ord}}_{\tilde{Z}}\Vert D(\beta )\Vert >0$. Then for any $\alpha \in [\beta ,m(\eta ,D))$ we have $${\mathrm{ord}}_{\tilde{Z}}\Vert D(\alpha )\Vert \ge \alpha -\beta +{\mathrm{ord}}_{\tilde{Z}}\Vert D(\beta )\Vert .(1)$$

In particular, if $T_X$ is nef, then the inequality Equation 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 $\delta$. Set $l:=k(\alpha -\beta )$. We may assume that $k$ is sufficiently large and sufficiently divisible so that $l$, and $l\delta$ are both integers. Note that a section $\sigma \in \mathrm{\Gamma }(X,{\mathcal{O}}_X(kL)\otimes {\mathfrak{m}}_x^{k\alpha })$ defines a morphism $${\mathcal{D}}_{kL}^l\otimes {\mathcal{O}}_X(l\delta L)\to {\mathcal{O}}_X((k+l\delta )L)\otimes {\mathcal{I}}_{{\mathrm{\Sigma }}_l}.$$ By the assumption that the sheaf ${\mathcal{D}}_{kL}^l\otimes {\mathcal{O}}_X(l\delta L)$ of differential operators of order $\le l$ is generated by its sections, we have a morphism of global sections $$\begin{array}{rl}{d}^l:H^0(X,{\mathcal{D}}_{kL}^l\otimes {\mathcal{O}}_X(l\delta L))& \to H^0(X,{\mathcal{O}}_X((k+l\delta )L))\\ D& \mapsto D(\sigma )\end{array}.$$ We call the sections in the image of the morphism $d^l$ differential sections of order $l$. Then $${\mathrm{mult}}_{x}D(\sigma )\ge k\alpha -k(\alpha -\alpha (Z))=k\alpha (Z)$$ for any differential operator $D$. Therefore, for any $D$, the differential section $D(\sigma )$ is also in $H^0(X,{\mathcal{O}}_X((k+l\delta )L)\otimes {\mathfrak{m}}_x^{k\alpha (Z)})$ and hence $${\mathrm{mult}}_{Z}D(\sigma )\ge k{\mathrm{ord}}_{\tilde{Z}}\Vert D(\beta )\Vert .$$ Assume that $\sigma$ is a general section and $x\in X$ is a general point such that ${\mathrm{mult}}_Z\sigma ={\mathrm{mult}}_x\sigma$. It can be checked that there exists a differential operator of order $l$ such that ${\mathrm{mult}}_{x}D(\sigma )={\mathrm{mult}}_x\sigma -l$. By upper semi continuity, we see that ${\mathrm{mult}}_{Z}D\le {\mathrm{mult}}_x\sigma -l$. Therefore, $$\begin{array}{rl}{\mathrm{ord}}_{\tilde{Z}}\Vert D(\alpha )\Vert =& {\mathrm{mult}}_x(\sigma )=l+{\mathrm{mult}}_x(D(\sigma ))\\ \ge & \alpha -\beta +{\mathrm{ord}}_{\tilde{Z}}\Vert D(\beta )\Vert \end{array}$$ 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 ${\mathcal{D}}_{kL}^l\otimes {\mathcal{O}}_X(l\delta L)$ of differential operators of order $\le l$ is generated by its sections. The proof is then completed.

2 Upper bounds for restricted volumes

Let $D$ be an ample integral divisor on $X$ and $f:Y\to X$ be the blow-up of $X$ at $\eta$ with the exceptional divisor $E$. The Seshadri constant of $L$ at $\eta$ is defined by $$\epsilon (\eta ,D)=\underset{\alpha \in \mathbb{Q}}{sup}\{\alpha \ge 0\mid f^{\ast }D-\alpha E\text{is nef}\}.$$

A subvariety $Z$ is Seshadri exceptional if it is the largest dimensional subvariety such that ${\left({\displaystyle \frac{D^{\dim Z}Z}{{\mathrm{mult}}_{\eta }Z}}\right)}^{\frac{1}{\dim Z}}=\epsilon (\eta ,D)$.

When $Z=C_{\eta }$ is an Seshadri exceptional curve, we have $\alpha (C_{\eta })=\epsilon (\eta ,D)$.

For any $\alpha <\epsilon (\eta ,D)$, we know that $\mathbf{B}(f^{\ast }D-\alpha E)$ is empty.
So $\alpha (C_{\eta })\ge \epsilon (\eta ,D)$. Assume that $\alpha (C_{\eta })>\epsilon (\eta ,D)$. Then there exists a number $\epsilon (\eta ,D)<\beta <\alpha$ such that $\widetilde{C_{\eta }}\notin \mathbf{B}(f^{\ast }D-\beta E)$. Consequently, $0<(f^{\ast }D-\beta E)\cdot \widetilde{(}{C}_{\eta })<(f^{\ast }D-\epsilon (\eta ,D)E)\cdot \widetilde{C_{\eta }}=0$. Therefore, $\alpha (C_{\eta })=\epsilon (\eta ,D)$.

We define $${\epsilon }_1=inf\{\alpha \in {\mathbf{Q}}_{\ge 0}\mid \dim (\mathbf{B}(f^{\ast }D-\alpha E))\cap E\ge 1\}.$$

It’s a non-trivial fact that $\epsilon (\eta ,D)\le {\epsilon }_1$ (see (Nakamaye 2000)).

An application of Lemma 2 and Seshadri constants is the following theorem on abelian threefold which was first proved by Lozovanu (Lozovanu 2018) using Newton-Okounkov bodies. We follow the argument in (Cascini and Nakamaye 2014).

Theorem 2 Let $X$ be a abelian threefold and $D$ be am ample divisor on $X$. Let $C\subset X$ be a curve with $q={\mathrm{mult}}_{\eta }X\ge 2$. Assume that $m(\eta ,D)>\alpha (C)$. Then $$\mathrm{Vol}(Y|E,D(\gamma )\le \{\begin{array}{ll}{\gamma }^2-q(\gamma -\alpha (C){)}^2& \text{when}\gamma \le {\displaystyle \frac{q}{q-1}}\alpha (C)\\ {\displaystyle \frac{q}{q-1}}\alpha (C{)}^2& \text{when}\gamma \ge {\displaystyle \frac{q}{q-1}}\alpha (C).\end{array}$$

Proof. If $C$ is not in $B(D(\gamma ))\cap E$, then the theorem follows from the proof of Proposition 3.2 in (Cascini and Nakamaye 2014) and Lemma @ref(lem:Nakamaye-generalization).

Suppose that $C$ is contained in $B(D(\gamma ))\cap E$. In $E={\mathbb{P}}^2$, taking away from $D(\gamma ){|}_E$ a curve whose degree is at least $(\gamma -\alpha (c))$, we find that $$\mathrm{Vol}(Y|E,D(\gamma )\le (\gamma -(\gamma -\alpha (C)){)}^2=\alpha (C{)}^2<\frac{q}{q-1}(\alpha (C){)}^2.$$

Note that ${\gamma }^2-q(\gamma -\alpha (C){)}^2$ reaches it maximum when $\gamma ={\displaystyle \frac{q}{q-1}}\alpha (C)$. The theorem then follows.

References

Cascini, Paolo, and Michael Nakamaye. 2014. “Seshadri Constants on Smooth Threefolds.” Adv. Geom. 14 (1): 59–79. https://doi.org/10.1515/advgeom-2013-0012.

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.

Lozovanu, Victor. 2018. “Singular Divisors and Syzygies of Polarized Abelian Threefolds.” ArXiv e-Prints, March. https://arxiv.org/abs/1803.08780.

Nakamaye, Michael. 2000. “Stable Base Loci of Linear Series.” Mathematische Annalen 318 (4): 837–47. https://doi.org/10.1007/s002080000149.

———. 2005. “Seshadri Constants at Very General Points.” Trans. Amer. Math. Soc. 357 (8): 3285–97. https://doi.org/10.1090/S0002-9947-04-03668-2.