Listeria monocytogenes isolates from Western Cape, South Africa exhibit resistance to multiple antibiotics and contradicts certain global resistance patterns

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.