Chaos and stability of a fractional model of the cyber ecosystem

1.   Introduction

Throughout the paper, we work over an algebraically closed field $\Bbbk$ of characteristic zero. Let $C$ be a nonsingular projective curve of genus $g\geq 0$, and $L$ be a very ample line bundle on $C$. The complete linear system $|L|$ embeds $C$ into a projective space $\mathbb{P}^r: = \mathbb{P}(H^0(C,L))$. For an integer $k\geq 0$, the $k$-th secant variety

of $C$ in $\mathbb{P}^r$ is the Zariski closure of the union of $(k+1)$-secant $k$-planes to $C$.

Assume that $\deg L\geq 2g+2k+1$. Then the $k$-th secant variety $\Sigma_k$ can be defined by using the secant sheaf $E_{k+1, L}$ and the secant bundle $B^k(L)$ as follows. Denote by $C_m$ the $m$-th symmetric product of $C$. Let

be the morphism sending $(\xi,x)$ to $\xi+x$, and $p \colon C_k \times C \rightarrow C$ the projection to $C$. The secant sheaf $E_{k+1,L}$ on $C_{k+1}$ associated to $L$ is defined by

which is a locally free sheaf of rank $k+1$. Notice that the fiber of $E_{k+1,L}$ over $\xi \in C_{k+1}$ can be identified with $H^0(\xi, L|_{\xi})$. The secant bundle of $k$-planes over $C_{k+1}$ is

equipped with the natural projection $\pi_k \colon B^{k}(L) \rightarrow C_{k+1}$. We say that a line bundle $\mathcal{L}$ on a variety $X$ separates $m+1$ points if the natural restriction map $H^0(X, \mathcal{L}) \to H^0(\xi, \mathcal{L}|_{\xi})$ is surjective for any effective zero-cycle $\xi \subseteq X$ with $\rm{length}(\xi) = m+1$. Notice that a line bundle $\mathcal{L}$ is globally generated if and only if $\mathcal{L}$ separates $1$ point, and $\mathcal{L}$ is very ample if and only if $\mathcal{L}$ separates $2$ points. Since $\deg L \geq 2g+k$, it follows from Riemann–Roch that $L$ separates $k+1$ points. Then the tautological bundle $\mathscr{O}_{B^k(L)}(1)$ is globally generated. We have natural identifications

and therefore, the complete linear system $| \mathscr{O}_{B^k(L)}(1)|$ induces a morphism

The $k$-th secant variety $\Sigma_k = \Sigma_k(C, L)$ of $C$ in $\mathbb{P}^r$ can be defined to be the image $\beta_k(B^k(L))$. Bertram proved that $\beta_k \colon B^k(L) \to \Sigma_k$ is a resolution of singularities (see [1,Section 1]).

It is clear that there are natural inclusions

The preimage of $\Sigma_{k-1}$ under the morphism $\beta_k$ is actually a divisor on $B^k(L)$. Thus there exits a natural morphism from $B^k(L)$ to the blowup of $\Sigma_k$ along $\Sigma_{k-1}$. Vermeire proved that $B^1(L)$ is indeed the blowup of $\Sigma_1$ along $\Sigma_0 = C$ ([3,Theorem 3.9]). In the recent work [2], we showed that $B^k(L)$ is the normalization of the blowup of $\Sigma_k$ along $\Sigma_{k-1}$ ([2,Proposition 5.13]), and raised the problem asking whether $B^k(L)$ is indeed the blowup itself ([2,Problem 6.1]). The purpose of this paper is to give an affirmative answer to this problem by proving the following:

Theorem 1.1. Let $C$ be a nonsingular projective curve of genus $g$, and $L$ be a line bundle on $C$. If $\deg L \geq 2g+2k+1$ for an integer $k \geq 1$, then the morphism $\beta_k \colon B^k(L) \to \Sigma_k(C, L)$ is the blowup of $\Sigma_k(C,L)$ along $\Sigma_{k-1}(C,L)$.

To prove the theorem, we utilize several line bundles defined on symmetric products of the curve. Let us recall the definitions here and refer the reader to [2] for further details. Let

be the $(k+1)$-fold ordinary product of the curve $C$, and $p_i \colon C^{k+1} \to C$ be the projection to the $i$-th component. The symmetric group $\mathfrak{{S}}_{k+1}$ acts on $p_1^*L \otimes \cdots\otimes p_{k+1}^*L$ in a natural way: a permutation $\mu\in \mathfrak{{S}}_k$ sends a local section $s_1\otimes \cdots \otimes s_{k+1}$ to $s_{\mu(1)}\otimes \cdots \otimes s_{\mu(k+1)}$. Then $p_1^*L \otimes \cdots\otimes p_{k+1}^*L$ is invariant under the action of $\mathfrak{{S}}_{k+1}$, so it descends to a line bundle $T_{k+1}(L)$ on the symmetric product $C_{k+1}$ via the quotient map $q \colon C^{k+1} \to C_{k+1}$. We have $q^*T_{k+1}(L) = p_1^*L \otimes \cdots\otimes p_{k+1}^*L$. Define a divisor $\delta_{k+1}$ on $C_{k+1}$ such that the associated line bundle $\mathscr{O}_{C_{k+1}}(\delta_{k+1}) = \det \big(\sigma_{k+1,*}( \mathscr{O}_{C_k\times C})\big)^*$. Let

be a line bundle on $C_{k+1}$. When $k = 0$, we use the convention that $T_1(L) = E_{1,L} = L$ and $\delta_1 = 0$.

The main ingredient in the proof of Theorem 1.1 is to study the positivity of the line bundle $A_{k+1,L}$. Some partial results and their geometric consequences have been discussed in [2,Lemma 5.12 and Proposition 5.13]. Along this direction, we establish the following proposition to give a full picture in a general result describing the positivity of the line bundle $A_{k+1,L}$. This may be of independent interest.

Proposition 1.2. Let $C$ be a nonsingular projective curve of genus $g$, and $L$ be a line bundle on $C$. If $\deg L \geq 2g+2k+\ell$ for integers $k, \ell \geq 0$, then the line bundle $A_{k+1,L}$ on $C_{k+1}$ separates $\ell+1$ points.

In particular, if $\deg L \geq 2g+2k$, then $A_{k+1, L}$ is globally generated, and if $\deg L \geq 2g+2k+1$, then $A_{k+1,L}$ is very ample.

2.   Proof of the main theorem

In this section, we prove Theorem 1.1. We begin with showing Proposition 1.2.

Proof of Proposition 1.2. We proceed by induction on $k$ and $\ell$. If $k = 0$, then $A_{1,L} = L$ and $\deg L\geq 2g+\ell$. It immediately follows from Riemann–Roch that $L$ separates $\ell+1$ points. If $\ell = 0$, then $\deg L \geq 2g+2k$. By [2,Lemma 5.12], $A_{k+1,L}$ separates 1 point.

Assume that $k\geq 1$ and $\ell \geq 1$. Let $z$ be a length $\ell+1$ zero-dimensional subscheme of $C_{k+1}$. We aim to show that the natural restriction map

is surjective. We can choose a point $p \in C$ such that $X_p$ contains a point in the support of $z$, where $X_p$ is the divisor on $C_{k+1}$ defined by the image of the morphism $C_k\rightarrow C_{k+1}$ sending $\xi$ to $\xi+p$. Let $y: = z\cap X_p$ be the scheme-theoretic intersection, and $\mathscr{I}_x: = ( \mathscr{I}_z: \mathscr{I}_{X_p})$, which defines a subscheme $x$ of $z$ in $C_{k+1}$, where $\mathscr{I}_z$ and $\mathscr{I}_{X_p}$ are ideal sheaves of $z$ and $X_p$ in $C_{k+1}$, respectively. We have the following commutative diagram

where all rows and columns are short exact sequences. By tensoring with $A_{k+1,L}$ and taking the global sections of last two rows, we obtain the commutative diagram with exact sequences

in which we use the fact that $H^1(A_{k+1,L}(-X_p)) = 0$ (see the proof of [2,Lemma 5.12]). Note that $A_{k+1,L}(-X_p) = A_{k+1, L(-p)}$ and $A_{k+1,L}|_{X_p}\cong A_{k,L(-2p)}$, where we identify $X_p = C_k$.

Since $\rm{length}(y) \leq \rm{length}(z) = \ell+1$ and $\deg L(-2p) \geq 2g+2(k-1)+\ell$, the induction hypothesis on $k$ implies that $r_{y,k,L(-2p)}$ is surjective. On the other hand, if $x = \emptyset$, which means that $z$ is a subscheme of $X_p$, then trivially $r_{x,k+1,L(-p)}$ is surjective. Otherwise, suppose that $x\neq \emptyset$. By the choice of $X_p$, we know that $y$ is not empty, and therefore, we have $\rm{length}(x) \leq \rm{length}(z)-1 = \ell$. Now, $\deg L(-p) \geq 2g + 2k + (\ell -1)$, so the induction hypothesis on $\ell$ implies that $L(-p)$ separates $\ell$ points. In particular, $r_{x,k+1,L(-p)}$ is surjective. Hence $r_{z,k+1,L}$ is surjective as desired.

Lemma 2.1. Let $\varphi \colon X \to Y$ be a finite surjective morphism between two varieties. If $\varphi^{-1}(q)$ is scheme theoretically a reduced point for each closed point $q \in Y$, then $\varphi$ is an isomorphism.

Proof. Note that $\varphi$ is proper, injective, and unramifield. Then it is indeed a classical result that $\varphi$ is an isomorphism. Here we give a short proof for reader's convenience. The problem is local. We may assume that $X = {\rm{Spec}}\; B$ and $Y = {\rm{Spec}}\; A$ for some rings $A,B$. We may regard $A$ as a subring of $B$. For any $q \in Y$, let $p: = \varphi^{-1}(q) \in X$. It is enough to show that the localizations $A': = A_{\mathfrak{m}_q}$ and $B': = B_{\mathfrak{m}_p}$ are isomorphic. Let $\mathfrak{m}_q', \mathfrak{m}_p'$ be the maximal ideals of the local rings $A',B'$, respectively. The assumption says that $\mathfrak{m}_q'B' = \mathfrak{m}_p'$. We have

By Nakayama lemma, we obtain $B'/A' = 0$.

We keep using the notations used in the introduction. Recall that $C$ is a nonsingular projective curve of genus $g \geq 0$, and $L$ is a very ample line bundle on $C$. Consider $\xi_{k} \in C_{k}$ and $x \in C$, and let $\xi: = \xi_{k} + x \in C_{k+1}$. The divisor $\xi_{k}$ spans a $k$-secant $(k-1)$-plane $\mathbb{P}(H^0(\xi_{k}, L|_{\xi_{k}}))$ to $C$ in $\mathbb{P}(H^0(C,L))$, and it is naturally embedded in the $(k+1)$-secant $k$-plane $\mathbb{P}(H^0(\xi,L|_{\xi}))$ spanned by $\xi$. This observation naturally induces a morphism

To see it in details, we refer to [1,p.432,line –5]. We define the relative secant variety $Z = Z_{k-1}$ of $(k-1)$-planes in $B^k(L)$ to be the image of the morphism $\alpha_{k,1}$. The relative secant variety $Z$ is a divisor in the secant bundle $B^k(L)$, and it is the preimage of $(k-1)$-th secant variety $\Sigma_{k-1}$ under the morphism $\beta_k$. It plays the role of transferring the codimension two situation $(\Sigma_k,\Sigma_{k-1})$ into the codimension one situation $(B^k(L),Z)$. We collect several properties of $Z$.

Proposition 2.2. ([2,Proposition 3.15,Theorem 5.2,and Proposition 5.13]) Recall the situation described in the diagram

Let $H$ be the pull back of a hyperplane divisor of $\mathbb{P}^r$ by $\beta_k$, and let $I_{\Sigma_{k-1}|\Sigma_k}$ be the ideal sheaf on $\Sigma_k$ defining the subvariety $\Sigma_{k-1}$. Then one has

1. $\mathscr{O}_{B^k(L)}((k+1)H-Z) = \pi^*_kA_{k+1,L}$.

2. $R^i\beta_{k,*} \mathscr{O}_{B^k(L)}(-Z) = \begin{cases} I_{\Sigma_{k-1}|\Sigma_k}& \mathit{if \; i = 0} \\ 0 & \mathit{if \; i >0}. \end{cases}$

3. $I_{\Sigma_{k-1}|\Sigma_k}\cdot \mathscr{O}_{B^k(L)} = \mathscr{O}_{B^k(L)}(-Z)$.

As a direct consequence of the above proposition, we have an identification

We are now ready to give the proof of Theorem 1.1.

Proof of Theorem 1.1. Let

be the blowup of $\Sigma_k$ along $\Sigma_{k-1}$ with exceptional divisor $E$. As $I_{\Sigma_{k-1}|\Sigma_k}\cdot \mathscr{O}_{B^k(L)} = \mathscr{O}_{B^k(L)}(-Z)$ (see Proposition 2.2), there exists a morphism $\alpha$ from $B^k(L)$ to the blowup $\widetilde{\Sigma}_k$ fitting into the following commutative diagram

We shall show that $\alpha$ is an isomorphism.

Write $V: = H^0(\Sigma_{k}, I_{\Sigma_{k-1}|\Sigma_k}(k+1))$. As proved in [2,Theorem 5.2], $I_{\Sigma_{k-1}|\Sigma_k}(k+1)$ is globally generated by $V$. This particularly implies that on the blowup $\widetilde{\Sigma}_k$ one has a surjective morphism $V\otimes \mathscr{O}_{\widetilde{\Sigma}_k}\rightarrow b^* \mathscr{O}_{\Sigma_{k}}(k+1)(-E)$, which induces a morphism

On the other hand, one has an identification $V = H^0(C_{k+1}, A_{k+1,L})$ by Proposition 2.2. Recall from Proposition 1.2 that $A_{k+1,L}$ is very ample. So the complete linear system $|V| = |A_{k+1,L}|$ on $C_{k+1}$ induces an embedding

Also note that $\alpha^*(b^* \mathscr{O}_{\Sigma_{k}}(k+1)(-E)) = \beta_k^* \mathscr{O}_{\Sigma_k}(k+1)(-Z) = \pi_k^*A_{k+1,L}$ by Proposition 2.2. Hence we obtain the following commutative diagram

Take an arbitrary closed point $x \in \widetilde{\Sigma}_k$, and consider its image $x': = b(x)$ on $\Sigma_{k}$. There is a nonnegative integer $m \leq k$ such that $x'\in \Sigma_m \setminus \Sigma_{m-1} \subseteq \Sigma_k$. In addition, the point $x'$ uniquely determines a degree $m+1$ divisor $\xi_{m+1,x'}$ on $C$ in such a way that $\xi_{m+1,x'} = \Lambda \cap C$, where $\Lambda$ is a unique $(m+1)$-secant $m$-plane to $C$ with $x' \in \Lambda$ (see [2,Definition 3.12]). By [2,Proposition 3.13], $\beta_k^{-1}(x') \cong C_{k-m}$ and $\pi_k(\beta_k^{-1}(x')) = \xi_{m+1,x'} + C_{k-m} \subseteq C_{k+1}$. Consider also $x'': = \gamma(x)$ which lies in the image of $\psi$. As $\psi$ is an embedding, we may think $x''$ as a point of $C_{k+1}$. Now, through forming fiber products, we see scheme-theoretically

However, the restriction of the morphism $\pi_k$ on $\beta_k^{-1}(x')$ gives an embedding of $C_{k-m}$ into $C_{k+1}$. This suggests that $\pi_k^{-1}(x'') \cap \beta_k^{-1}(x')$ is indeed a single reduced point, and so is $\alpha^{-1}(x)$. Finally by Lemma 2.1, $\alpha$ is an isomorphism as desired.