On the first nontrivial strand of syzygies of projective schemes and condition \(\mathrm{ND}(\ell)\) (Q6093188)

Let $X \subseteq \mathbb{P}^{n+e}$ be a non-degenerate closed subscheme of dimesnion $n$ and codimension $e$ defined over an algebraically closed field $\mathbf{k}$. Although some results of the paper under review hold in this generality, we assume that $X$ is a projective variety and the characteristic of the base field $\mathbf{k}$ is zero for convenience. After Green's pioneering work on syzygies, there has been a great deal of interest in understanding the Betti tables of projective varieties. The Betti table of $X \subseteq \mathbb{P}^{n+e}$ consists of the graded Betti numbers \[ \beta_{i,j}(X):= \dim_{\mathbf{k}} \operatorname{Tor}_i^R(R/I_{X|\mathbb{P}^r}, \mathbf{k})_{i+j}, \] where $R:=\mathbf{k}[x_0, \ldots, x_{n+e}]$ is the homogeneous coordinate ring of $\mathbb{P}^{n+e}$. Previously, Han-Kwak proved that \[ \beta_{i,1}(X) \leq i \binom{e+1}{ i+1}\text{ for }i \geq 1 \] and the equality holds for some (or each) $1 \leq i \leq e$ if and only if $X \subseteq \mathbb{P}^{n+e}$ is arithmetically Cohen-Macaulay with $2$-linear resolution. One may expect to generalize this result to the first nontrivial strand of the Betti table -- assuming $H^0(\mathbb{P}^{n+e}, \mathscr{I}_{X|\mathbb{P}^{n+e}}(\ell)) = 0$ for $\ell \geq 1$, we seek to find a reasonable upper bound for $\beta_{i, \ell}(X)$. For this purpose, we need an additional condition that is $\operatorname{ND}(\ell)$ condition introduced in this paper. We say that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(\ell)$ condition if $H^0(\Lambda, \mathscr{I}_{X \cap \Lambda|\Lambda}(\ell)) = 0$ for every general linear subspace $\Lambda \subseteq \mathbb{P}^{n+e}$ with $\dim \Lambda \geq e$. It is equivalent to $\operatorname{Gin}(I_{X|\mathbb{P}^{n+e}}) \subseteq (x_0, \ldots, x_{e-1})^{\ell+1}$, where $\operatorname{Gin}(I_{X|\mathbb{P}^{n+e}})$ is the generic initial ideal with respect to the degree reverse lexicographic order (Proposition 2.3). Note that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(1)$ condition if and only if $X \subseteq \mathbb{P}^{n+e}$ is non-degenerate. This means that $\operatorname{ND}(1)$ condition is automatic while $\operatorname{ND}(\ell)$ condition for $\ell \geq 2$ is nontrivial. In Section 4, relevant examples and some questions on $\operatorname{ND}(\ell)$ condition are presented. The first main result of the paper is Theorem 1.1: If $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(\ell)$ condition, then \[ \beta_{i, \ell}(X) \leq \binom{i+\ell-1}{ \ell} \binom{e + \ell}{i + \ell}\text{ for }i \geq 1 \] and the equality holds for some (or each) $1 \leq i \leq e$ if and only if $X \subseteq \mathbb{P}^{n+e}$ is arithmetically Cohen-Macaulay with $(\ell+1)$-linear resolution. Next, recall that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{d,p}$ condition if $\beta_{i,j}(X) = 0$ for $i \leq p$ and $j \geq d$. A well-known result of Eisenbud-Green-Hulek-Popescu says that if $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{2,e}$, then $X$ is arithmetically Cohen-Macaulay with $2$-linear resolution. To generalized this result, we also need $\operatorname{ND}(\ell)$ condition. More precisely, the second main result of the paper (Theorem 1.2) states ``if $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{d,e}$ condition and $\operatorname{ND}(d-1)$ condition, then $X$ is arithmetically Cohen-Macaulay with $d$-linear resolution.''