Higher syzygies on general polarized abelian varieties of type $(1,\dots,1,d)$

Throughout this paper, we work over the complex number field $\mathbb{C}$.

For an ample line bundle $L$ on an abelian variety $X$ of dimension $g$, we can associate a sequence of positive integers $(d_{1},\dots,d_{g})$ with $d_{1}|d_{2}|\cdots|d_{g}$, called the type of $L$. It is well known that $L$ is basepoint free if $d_{1}\geqslant 2$ and projectively normal if $d_{1}\geqslant 3$. On the other hand, in the case $d_{1}=1$, equivalently the case when $L$ is not written as some multiple of another line bundle, basepoint freeness or projective normality of $L$ is more subtle. In [DHS94], the authors investigate general $L$ of type $(1,\dots,1,d)$ and prove the following theorem.

On the other hand, L. Fuentes García investigates projective normality based on the work [Iye03] of J. N. Iyer, and conjectures the following:

For $g=2$, this conjecture follows from [Laz90], [FG04] or [Iye99]. Fuentes García proves this conjecture for $g=3,4$ using results in [Iye03] and some calculations of the ranks of suitable matrices using computer. We note that $d=h^{0}(L)\geqslant 2^{g+1}-1$ is a necessary condition for the projective normality of $L$ since $\dim\operatorname{Sym}^{2}H^{0}(L)\geqslant\dim H^{0}(L^{\otimes 2})$ must hold for such $L$. Hence § 1 states that it is a sufficient condition as well for general $(X,L)$.

In this paper, we prove § 1 affirmatively. In fact, we prove not only projective normality but also property $(N_{p})$ as follows:

We refer the readers to [Laz04, Chapter 1.8.D], [Eis05] for the definition of property $(N_{p})$. We just note here that $(N_{p})$’s consist an increasing sequence of positivity properties. For example, ($N_{0}$) holds for $L$ if and only if $L$ defines a projectively normal embedding, and ($N_{1}$) holds if and only if ($N_{0}$) holds and the homogeneous ideal of the embedding is generated by quadrics. Usually $(N_{p})$ is considered for $p\geqslant 0$, but we add the basepoint freeness in the sequence of positivity properties as $(N_{-1})$, as in [Loz18], [Jia21]. Hence the case $p=-1$ of Theorem 1.2 recovers Theorem 1.1 (1) and the case $p=0$ of Theorem 1.2 proves § 1 affirmatively.

For an abelian surface $X$, it is known that a very ample line bundle $L$ of type $(1,d)$ with $d\geqslant 7$ is projectively normal and the homogeneous ideal of $X$ embedded by $|L|$ is generated by quadrics and cubics by [Laz90], [FG04], [Ago17]. The last statement of Theorem 1.2 generalizes this result to higher dimensions at least for general $(X,L)$. Furthermore, Theorem 1.2 gives a better bound of $d$ than the bounds obtained by [KL19], [Ito18] in dimension two and [Iye03], [LPP11], [Ito20], [Jia21] in higher dimensions. See Remarks 4, 5 for details.

In the rest of Introduction, we explain the idea of the proof. In [JP20], Z. Jiang and G. Pareschi introduce an invariant $\beta(l)=\beta(X,l)\in(0,1]$, called the basepoint-freeness threshold, for the class $l\in\operatorname{Pic}X/\operatorname{Pic}^{0}X$ of an ample line bundle $L$. By [JP20], [Cau20a], a suitable upper bound of $\beta(l)$ implies property $(N_{p})$ as follows:

Hence the following theorem implies Theorem 1.2.

In [Ito20], the author observes a similarity between $\beta(l)^{-1}$ and Seshadri constants. Since Seshadri constants are lower-semicontinuous, it is natural to ask whether $\beta(l)$ is upper-semicontinuous or not. In fact, this is the case as we see in § 3. Hence, Theorem 1.4 is reduced to finding an example $(X_{0},L_{0})$ of type $(1,\dots,1,d)$ such that $\beta(l_{0})\leqslant 1/m$ or $<1/m$.

Theorem 1.1 (3) is proved in [DHS94] by degenerating polarized abelian varieties to a polarized variety $(X^{\prime}_{0},L^{\prime}_{0})$ whose normalization is a $(\mathbb{P}^{1})^{g-1}$-bundle over an elliptic curve and showing that $L^{\prime}_{0}$ is very ample. Contrary to very ampleness, $\beta(l)$ is defined only for abelian varieties. Hence we do not use such degenerations but find $(X_{0},L_{0})$ as a polarized abelian variety. In fact, we construct such $(X_{0},L_{0})$ as a suitable polarization on a product of elliptic curves.

We note that Theorem 1.3 is also used to show $(N_{p})$ in [Ito20], [Jia21], where techniques to cutting minimal log canonical centers are used to bound $\beta(l)$ from above. In this paper, we do not need such techniques.

This paper is organized as follows. In § 2, we recall some notation. In § 3, we show the upper-semicontinuity of $\beta(l)$. In § 4, we study $\beta(l)$ of polarized abelian surfaces and show Theorem 1.4 for $g=2$. In § 5, we prove Theorems 1.2, 1.4 in any dimension. In Appendix, we compute $\beta(l)$ of general polarized abelian surfaces $(X,l)$ of type $(1,d)$ for some $d$.

Let $X$ be an abelian variety of dimension $g$. We denote the origin of $X$ by $o_{X}$ or $o\in X$. For $b\in\mathbb{Z}$, we denote the multiplication-by-$b$ isogeny by

For an ample line bundle $L$ on $X$, we call $(X,L)$ or $(X,l)$ a polarized abelian variety, where $l\in\mathrm{NS}(X)=\operatorname{Pic}(X)/\operatorname{Pic}^{0}(X)$ is the class of $L$ in the Neron-Severi group of $X$. Let

where $t_{p}:X\rightarrow X$ is the translation by $p$ on $X$. It is known that there exist positive integers $d_{1}|d_{2}|\cdots|d_{g}$ such that $K(L)\simeq(\bigoplus_{i=1}^{g}\mathbb{Z}/d_{i}\mathbb{Z})^{\oplus 2}$ as abelian groups. The vector $(d_{1},\dots,d_{g})$ is called the type of $L$. Since $K(L)$ depends only on the class $l$ of $L$, $(d_{1},\dots,d_{g})$ is called the type of $l$ as well. It is known that $\chi(l)=\prod_{i=1}^{g}d_{i}$ holds.

For a coherent sheaf $\mathcal{F}$ on $X$ and $x\in\mathbb{Q}$, a $\mathbb{Q}$-twisted coherent sheaf $\mathcal{F}\langle xl\rangle$ is the equivalence class of the pair $(\mathcal{F},xl)$, where the equivalence is defined by

for any line bundle $L$ representing $l$ and any $m\in\mathbb{Z}$.

Recall some notions of generic vanishing: a coherent sheaf $\mathcal{F}$ on $X$ is said to be IT(0) if $h^{i}(X,\mathcal{F}\otimes P_{\alpha})=0$ for any $i>0$ and any $\alpha\in\widehat{X}=\operatorname{Pic}^{0}(X)$, where $P_{\alpha}$ is the algebraically trivial line bundle on $X$ corresponding to $\alpha$. It is said to be GV if $\operatorname{codim}_{\widehat{X}}\{\alpha\in\widehat{X}\,|\,h^{i}(X,\mathcal{F}\otimes P_{\alpha})>0\}\geqslant i$ for any $i>0$.

In [JP20], such notions are extended to the $\mathbb{Q}$-twisted setting. A $\mathbb{Q}$-twisted coherent sheaf $\mathcal{F}\langle xl\rangle$ for $x=a/b$ with $b>0$ is said to be IT(0) or GV if so is $\mu_{b}^{*}\mathcal{F}\otimes L^{ab}$. We note that this definition does not depend on the representation $x=a/b$ nor the choice of $L$ representing $l$. By [JP20, Theorem 5.2], $\mathcal{F}\langle xl\rangle$ is GV if and only if $\mathcal{F}\langle(x+x^{\prime})l\rangle$ is IT(0) for any rational number $x^{\prime}>0$.

In [JP20], an invariant $0<\beta(l)\leqslant 1$ is introduced for a polarized abelian variety $(X,l)$. It is defined using cohomological rank functions, which are also defined in [JP20], but $\beta(l)$ is characterized by the notion IT(0) as follows:

We use the following lemma to estimate $\beta(l)$.

In this section, we prove the upper-semicontinuity of $\beta(l)$ as follows:

Let $k\geqslant 1$ be an integer and take an isogeny $f:E\rightarrow E^{\prime}$ between elliptic curves with $\ker f\simeq\mathbb{Z}/k\mathbb{Z}$, e.g. we take $f:\mathbb{C}/(\mathbb{Z}+k\tau\mathbb{Z})\rightarrow\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ induced from $\mathbb{Z}+k\tau\mathbb{Z}\subset\mathbb{Z}+\tau\mathbb{Z}$. For the dual isogeny $\hat{f}:E^{\prime}=\widehat{E^{\prime}}\rightarrow\widehat{E}=E$, it is well-known that $\hat{f}\circ f=\mu^{E}_{k}$ and $f\circ\hat{f}=\mu^{E^{\prime}}_{k}$ hold (see e.g. [Sil09, Chapter III, Theorem 6.2]).

Set $X=E\times E^{\prime}$ and

Now we can show Theorem 1.4 for abelian surfaces:

As stated in Introduction, a very ample line bundle $L$ of type $(1,d)$ with $d\geqslant 7$ on an abelian surface $X$ satisfies $(N_{0})$, that is, $L$ is projectively normal by [Laz90], [FG04]. Furthermore, in this case the homogeneous ideal of $X$ embedded by $|L|$ is generated by quadrics and cubics by [Ago17]. For $d\geqslant 10$, a general polarized abelian surface of type $(1,d)$ satisfies $(N_{1})$, that is, the homogeneous ideal of $X$ embedded by $|L|$ is generated by quadrics by [GP98].

By § 4, we can show Theorem 1.2 for abelian surfaces, which partially recovers and generalizes the above results to higher syzygies. Partial means that we cannot give explicit conditions for the generality of $(X,L)$ and the bound $d\geqslant 13$ for $(N_{1})$ is larger than the bound $d\geqslant 10$ in [GP98].

In this section, we prove Theorem 1.4. Although the argument becomes a little complicated, the essential idea is the same as the surface case. In the following examples, it is not difficult to bound $\beta(l)$. To show that the type is of the form $(1,\dots,1,d)$, we use § 4.

Let $g\geqslant 2,k_{1},\dots,k_{g-1}\geqslant 1$ be integers and set $k_{g}:=1$. Let $E_{g}$ be an elliptic curve and take an elliptic curve $E_{i}$ and an isogeny $f_{i}:E_{i}\rightarrow E_{g}$ for each $1\leqslant i\leqslant g-1$ with $\ker f_{i}\simeq\mathbb{Z}/k_{i}\mathbb{Z}$. Then we have $\hat{f}_{i}\circ f_{i}=\mu^{E_{i}}_{k_{i}},f_{i}\circ\hat{f}_{i}=\mu^{E_{g}}_{k_{i}}$ for the dual isogeny $\hat{f}_{i}:E_{g}\rightarrow E_{i}$. Let $X=E_{1}\times E_{2}\times\dots\times E_{g}$ and let $F_{i}=\operatorname{pr}_{i}^{*}(o_{E_{i}})$, where $\operatorname{pr}_{i}:X\rightarrow E_{i}$ is the projection to the $i$-th factor. A divisor $\Gamma$ on $X$ is defined by

The following proposition is a generalization of § 4 (ii), (iii) to higher dimensions: