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 .

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 $$\mathbf{B}(D):=\bigcap_{k} \mathrm{Bs}{\vert}kD{\vert}$$ the stable base locus of $$D$$, that is the intersection of the base loci of the linear systems $${\vert}kD{\vert}$$ 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 $$\mathrm{ord}_Z\lVert D\rVert$$ 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 $$k$$.

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

Restricted volumes have the following properties.

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

1. $$\mathrm{Vol}(X{\vert}Z, D)>0$$ if $$Z$$ is not contained in $$\mathbf{B}_+(D)$$.
2. $$\mathrm{Vol}(X{\vert}Z, D)$$ is numerical invariant.
3. for $$k$$ sufficiently large and divisible, and for any general elements $$D_{k,1}, \dots, D_{k,d}$$ in $$\lvert kD \rvert$$, we have $\mathrm{Vol}(X{\vert}Z, D)=\dfrac{\#(Z\cap D_{k,1}\cap \cdots\cap D_{k,d}\setminus \mathbf{B}(D))}{k^d},$ where $$d=\dim Z$$.
4. Let $$f: X'\to X$$ be a proper and birational morphism and $$Z'$$ be an irreducible subvariety on $$X'$$ such that $$f(Z')=Z$$. Then $\mathrm{Vol}(X'{\vert}Z', f^*D)=\mathrm{Vol}(X{\vert}Z, D).$
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 $\mathrm{Vol}(Y, D_\alpha)=\mathrm{Vol}(X, D)-n\int_0^\alpha \mathrm{Vol}(Y{\vert}E, D(\gamma))\mathrm{d} \gamma,$ where $$D(\gamma)=f^*D-\gamma E$$.

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 $D(\alpha)=f^*D-\alpha E,$ $\alpha(Z)=\inf_{\beta\in\mathbb{Q}}\{\tilde{Z}\subseteq \mathbf{B}(D(\beta) ) \},$ and $m(\eta,D)=\sup_{\alpha\in \mathbb{Q}_{\geq 0}}\{D(\alpha) ~\text{is \mathbb{Q}--effective}\}.$

When estimating the restricted volume $$\mathrm{Vol}(Y{\vert}E, D(\gamma))$$, 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 $$\D_{kL}^l\otimes\O_X(l\delta L)$$ of differential operators of order $$\le l$$ is generated by its sections for a rational number $$\delta$$, and 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 $$$\ord_{\widetilde{Z}}\lVert D(\alpha) \rVert\geq \alpha-\beta+\ord_{\widetilde{Z}}\lVert D(\beta) \rVert. \tag{1}$$$

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 $$\delta$$. 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 $$\sigma\in \Gamma(X, \O_X(kL)\otimes \mfm_x^{k\alpha})$$ defines a morphism $\D^l_{kL}\otimes\O_X(l\delta L)\to \O_X((k+l\delta)L) \otimes \mathcal{I}_{\Sigma_l}.$ 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))\\ D&\mapsto D(\sigma) \end{aligned}. We call the sections in the image of the morphism $$d^l$$ differential sections of order $$l$$. Then $\mult_xD(\sigma)\ge k\alpha-k(\alpha-\alpha(Z))=k\alpha(Z)$ for any differential operator $$D$$. Therefore, for any $$D$$, the differential section $$D(\sigma)$$ is also in $$H^0(X, \O_X((k+l\delta)L)\otimes\mfm_x^{k\alpha(Z)})$$ 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=&\mult_x(\sigma)=l+\mult_x(D(\sigma))\\ \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 , 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.

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 $\varepsilon(\eta, D)=\sup_{\alpha\in \mathbb{Q}}\{\alpha\geq 0 \mid f^*D-\alpha E ~\text{is nef}\}.$

A subvariety $$Z$$ is Seshadri exceptional if it is the largest dimensional subvariety such that $$\left(\dfrac{D^{\dim Z}Z}{\mult_\eta Z}\right)^{\frac{1}{\dim Z}}=\varepsilon(\eta, D)$$.

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

For any $$\alpha<\varepsilon(\eta, D)$$, we know that $$\mathbf{B}(f^*D-\alpha E)$$ is empty.
So $$\alpha(C_\eta)\geq \varepsilon(\eta, D)$$. Assume that $$\alpha(C_\eta)>\varepsilon(\eta, D)$$. Then there exists a number $$\varepsilon(\eta, D)<\beta<\alpha$$ such that $$\widetilde{C_\eta}\not\in\mathbf{B}(f^*D-\beta E)$$. Consequently, $$0<(f^*D-\beta E)\cdot \widetilde(C_\eta)< (f^*D-\varepsilon(\eta, D) E)\cdot \widetilde{C_\eta}=0$$. Therefore, $$\alpha(C_\eta)=\varepsilon(\eta, D)$$.

We define $\varepsilon_1=\inf\{\alpha\in \mathbf{Q}_{\geq 0}\mid \dim(\mathbf{B}(f^*D-\alpha E))\cap E\geq 1\}.$

It’s a non-trivial fact that $$\varepsilon(\eta, D)\leq \varepsilon_1$$ (see ).

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

Theorem 2 Let $$X$$ be a abelian threefold and $$D$$ be am ample divisor on $$X$$. Let $$C\subset X$$ be a curve with $$q=\mult_\eta X\geq 2$$. Assume that $$m(\eta, D)>\alpha(C)$$. Then $\mathrm{Vol}(Y{\vert}E, D(\gamma)\leq \begin{cases} \gamma^2-q(\gamma-\alpha(C))^2 & \text{when}~ \gamma\leq \dfrac{q}{q-1}\alpha(C) \\[1em] \dfrac{q}{q-1}\alpha(C)^2 & \text{when}~ \gamma\geq \dfrac{q}{q-1}\alpha(C). \end{cases}$

Proof. If $$C$$ is not in $$B(D(\gamma))\cap E$$, then the theorem follows from the proof of Proposition 3.2 in and Lemma 2.

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

Note that $$\gamma^2-q(\gamma-\alpha(C))^2$$ reaches it maximum when $$\gamma=\dfrac{q}{q-1}\alpha(C)$$. The theorem then follows.
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. http://arxiv.org/abs/1803.08780.
Nakamaye, Michael. 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.

Author's bio

Fei Ye (https://yfei.page) is an assistant professor at QCC-CUNY.

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Citation

Fei Ye (2020). Seshadri Constants and Restricted Volumes. Fei Ye's Math Blogs. /post/2020/11/22/seshardi-volume/
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