An invariant for embedded Fano manifolds covered by linear spaces

TL;DR

This paper introduces a new invariant for embedded Fano manifolds related to linear spaces and aims to classify those with large invariant values.

Contribution

It proposes a novel invariant $S_X$ for embedded Fano manifolds and provides a classification for manifolds with large $S_X$ values.

Findings

Defined the invariant $S_X$ for embedded Fano manifolds.

Classified Fano manifolds with large $S_X$ values.

Connected the invariant to the geometry of covering linear spaces.

Abstract

For an embedded Fano manifold $X$, we introduce a new invariant $S_X$ related to the dimension of covering linear spaces. The aim of this paper is to classify Fano manifolds $X$ which have large $S_X$.

Equations

$$X\models H_1\models H_2\models \cdots \models H_m$$

$$X\models {\mathbb{P}}^{n-1}\models {\mathbb{P}}^{n-2}\models \cdots \models {\mathbb{P}}^1\models \text{pt},$$

$$X\models Q^{n-2}\models Q^{n-4}\models \cdots \models \{\begin{array}{ll}{Q}^2\models \text{pt}& \text{ if }n\text{ is even}\\ Q^3\models Q^1& \text{ if }n\text{ is odd}\end{array},$$

$$X\models {\mathbb{P}}^1\times {\mathbb{P}}^{m-1}\models \{\begin{array}{l}\text{pt}\\ {\mathbb{P}}^{m-2}\models {\mathbb{P}}^{m-3}\models \cdots \models \text{pt}\end{array},$$

$$X\models \mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^1}(2)\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(1{)}^{m-1})(\cong {\text{Bl}}_{{\mathbb{P}}^{m-2}}{\mathbb{P}}^m)\models {\mathbb{P}}^{m-2}\models {\mathbb{P}}^{m-3}\models \cdots \models \text{pt},$$

$$X(=:H_0)\models H_1\models \cdots \models H_m$$

$$n_0-n_1\le (2m+1)-2(m-1)=3.$$

$$2m+1\ge 2(i-1)+n_{i-1}\ge 2(i-1)+2(m-i)+4=2m+2,$$

$$2m+1\ge 4+2(i-3)+n_{i-2}\ge 4+2(i-3)+2(2+m-i)=2m+2,$$

Full text

1. Introduction

We consider an embedded Fano manifold $X$, i.e., a smooth complex projective variety $X\subset\mathbb{P}^{N}$ with ample anti-canonical line bundle $-K_{X}$. If $X$ is covered by lines, then we consider a family $H_{1}$ of lines on $X$ through a fixed general point $x$. We can regard $H_{1}$ as a subvariety of $\mathbb{P}(T_{x}X^{\vee})$. If $H_{1}$ is also a Fano manifold covered by lines, then we can consider a family $H_{2}$ of lines on $H_{1}$ through a fixed general point. In this way, we inductively define an $m$-th order family $H_{m}$ of lines. In addition, we define an invariant $S_{X}$ as the greatest number $m$ such that there is an $m$-th order family of lines. As shown later, any Fano manifold $X$ satisfies $S_{X}\leq{\rm dim}\,X$, and $X$ is covered by linear spaces of dimension at least $S_{X}$. In particular, $S_{X}={\rm dim}\,X$ holds if and only if $X$ is a linear space.

The aim of this paper is to classify Fano manifolds $X$ of Picard number $1$ which have large $S_{X}$. We will prove the following theorem:

Theorem 1.1**.**

Let $X\subset\mathbb{P}^{N}$ be a Fano manifold of Picard number $1$ and dimension $n\geq 2$.

1. (1)

If $S_{X}>\frac{n}{2}$, then $X$ is a linear space $\mathbb{P}^{n}$. 2. (2)

If $S_{X}=\frac{n}{2}$, then $X$ is isomorphic to either a quadric hypersurface $Q^{2m}$ or the Grassmannian $G(2,\mathbb{C}^{m+2})$ $(n=2m\geq 4)$. 3. (3)

Assume that Conjecture 1.2 is true. If $S_{X}=\frac{n-1}{2}$, then $X$ is isomorphic to one of the following:

• (a)

a quadric hypersurface $Q^{2m+1}$ $(n=2m+1)$;

• (b)

the symplectic Grassmannian $SG(2,\mathbb{C}^{m+3})$ $(n=2m+1)$;

• (c)

a cubic hypersurface in $\mathbb{P}^{4}$ $(n=3)$;

• (d)

a complete intersection of two quadric hypersurfaces in $\mathbb{P}^{5}$ $(n=3)$;

• (e)

a $3$-dimensional linear section of the Grassmannian $G(2,\mathbb{C}^{5})$ in $\mathbb{P}^{9}$ $(n=3)$.

Conjecture 1.2**.**

Let $X\subset\mathbb{P}^{N}$ be a Fano manifold of Picard number $1$ which is covered by lines. Assume that a family of lines on $X$ through $x$ embedded in $\mathbb{P}(T_{x}X^{\vee})$ is projectively equivalent to $\mathbb{P}(\mathscr{O}_{\mathbb{P}^{1}}(2)\oplus\mathscr{O}_{\mathbb{P}^{1}}(1)^{m-1})$ embedded in $\mathbb{P}^{2m}$ by the tautological line bundle for a general point $x\in X$. Then $X$ is isomorphic to the symplectic Grassmannian $SG(2,\mathbb{C}^{m+3})$.

Projective $n$-folds covered by linear spaces of large dimension $m$ have been investigated by several authors. E. Sato studied the case $m\geq\frac{n}{2}$. We immediately obtain the first two arguments (1) and (2) of Theorem 1.1 from the following theorem:

Theorem 1.3** ([13], see also [11, Corollary 5.3]).**

Let $X$ be a smooth projective $n$-fold $X\subset\mathbb{P}^{N}$. Assume that $X$ is covered by linear spaces of dimension $m\geq\frac{n}{2}$. Then $X$ is one of the following:

• (i)

a linear $\mathbb{P}^{k}$-bundle $(k\geq m)$;

• (ii)

a quadric hypersurface $Q^{2m}$ $(m=\frac{n}{2})$;

• (iii)

the Grassmannian $G(2,\mathbb{C}^{m+2})$ $(m=\frac{n}{2})$.

C. Novelli and G. Occhetta classified the case $m=\frac{n-1}{2}$ under the assumption that the normal bundles of the covering linear spaces are numerically effective [11, Theorem 1.1]. However, the complete classification of the case $m=\frac{n-1}{2}$ is not known. So, the argument (3) of Theorem 1.1 is the main part of this paper.

In order to classify the case $S_{X}=\frac{n-1}{2}$, we need Conjecture 1.2, i.e., a characterization of sympletic Grassmannians in terms of families of lines. G. Occhetta, L. E. Solá Conde, and K. Watanabe’s recent result [12] says that Conjecture 1.2 is ture if the assumption holds for every point $x\in X$.

This paper is organized as follows. In section 2, we mention several facts concerning families of lines. In section 3, we introduce higher order families of lines and the invariant $S_{X}$. In section 4, we prove (3) of Theorem 1.1.

Acknowledgements**.**

The author would like to express his gratitude to his supervisor Professor Hajime Kaji for valuable discussions. The author would also like to thank Professor Kiwamu Watanabe for beneficial suggestions and comments. The author is also grateful to Professor Yasunari Nagai and Professor Daizo Ishikawa for useful comments.

2. Preliminaries

We denote by $\rho_{X}$ the Picard number of a manifold $X$. We refer to [10, I and II] for basic theory of families of rational curves. Throughout this paper, we consider a smooth projective Fano $n$-fold $X$ embedded in a projective space $\mathbb{P}^{N}$. Let $x$ be a fixed general point of $X$. We denote by $F_{1}(X,x)$ the Hilbert scheme of lines on $X$ passing through $x$. From now on, let $H$ be an irreducible component of $F_{1}(X,x)$, which is called a family of lines through $x$.

Lemma 2.1** ([6, Theorem 1.3 and Proposition 1.5]).**

We can regard $H$ as a smooth projective subvariety of the projectivised tangent space $\mathbb{P}(T_{x}X^{\vee})$ which has dimension $(-K_{X}\cdot H)-2$, where $\mathbb{P}(T_{x}X^{\vee}):=(T_{x}X\setminus\{0\})/\mathbb{C}^{\times}\cong\mathbb{P}^{n-1}$, and $(-K_{X}\cdot H)$ means the anti-canonical degree of a line parametrized by $H$.

Example 2.2** (see [6, 1.4]).**

1. (1)

If $X$ is a linear space $\mathbb{P}^{n}$, then $H$ is equal to $\mathbb{P}(T_{x}X^{\vee})$. 2. (2)

If $X$ is a quadric hypersurface $Q^{n}$ in $\mathbb{P}^{n+1}$ ($n\geq 3$), then $H$ is a quadric hypersurface $Q^{n-2}$ in $\mathbb{P}(T_{x}X^{\vee})$. 3. (3)

If $X$ is $\mathbb{P}^{k}\times\mathbb{P}^{m}$ (Segre embedding), then $H$ is a linear space $\mathbb{P}^{k-1}$ or $\mathbb{P}^{m-1}$ in $\mathbb{P}(T_{x}X^{\vee})$. 4. (4)

If $X$ is the Grassmannian $G(k,\mathbb{C}^{m})$ (Plücker embedding), then $H$ is $\mathbb{P}^{k-1}\times\mathbb{P}^{m-k-1}$ embedded in $\mathbb{P}(T_{x}X^{\vee})$ by the Segre embedding. 5. (5)

If $X$ is the symplectic Grassmannian $SG(k,\mathbb{C}^{m})$, then $H$ is $\mathbb{P}(\mathscr{O}_{\mathbb{P}^{k-1}}(2)\oplus\mathscr{O}_{\mathbb{P}^{k-1}}(1)^{m-2k})$ embedded in $\mathbb{P}(T_{x}X^{\vee})$ by the tautological line bundle. 6. (6)

If $X$ is a complete intersection of degrees $(d_{1},\ldots,d_{m})$ in $\mathbb{P}^{N}$ ($N\geq\sum d_{i}+2$), then $H$ is a complete intersection of degrees $(2,3,\ldots,d_{1},\ldots,2,3,\ldots,d_{m})$ in $\mathbb{P}(T_{x}X^{\vee})$.

Lemma 2.3**.**

If $H$ contains a linear space $L$ of dimension $m$, then $X$ contains a linear space of dimension $m+1$ through $x$.

Proof.

Let $U$ be the universal family of $H$, and let $\pi:U\rightarrow H$ and $e:U\rightarrow X$ be the associated morphisms. By the proof of [1, Lemma 2.3], we have that the restriction of $e$ to $\pi^{-1}(L)$ is the blow-up of $\mathbb{P}^{m+1}$ and $e(\pi^{-1}(L))$ is a linear space of dimension $m+1$ through $x$. ∎

Lemma 2.4**.**

We have the following:

1. (1)

*If ${\rm dim}\,H=n-1$ *$($i.e., $H=\mathbb{P}(T_{x}X^{\vee}))$, then $X$ is a linear space $\mathbb{P}^{n}$. 2. (2)

If $\rho_{X}=1$ and ${\rm dim}\,H=n-2$, then $X$ is a quadric hypersurface ${Q}^{n}$ $(n\geq 3)$. 3. (3)

If $\rho_{X}=1$ and ${\rm dim}\,H=n-3$, then $X$ is one of the following:

• (i)

a cubic hypersurface;

• (ii)

a complete intersection of two quadric hypersurfaces;

• (iii)

the Grassmannian $G(2,\mathbb{C}^{5})$ or a linear section of it.

Proof.

(1) follows from Lemma 2.3. If $\rho_{X}=1$, then $-K_{X}$ is linearly equivalent to $\mathscr{O}_{\mathbb{P}^{N}}({\rm dim}\,H+2)|_{X}$ by Lemma 2.1. So, (2) follows from [9], and (3) follows from [4] and [5]. ∎

Lemma 2.5**.**

If $\rho_{X}=1$ and ${\rm dim}\,H\geq\frac{n-1}{2}$, then $H\subset\mathbb{P}(T_{x}X^{\vee})$ is non-degenerate.

Proof.

By [6, Theorem 1.3 and Proposition 1.5], $H$ is contained in a smooth projective subvariety $V$ of $\mathbb{P}(T_{x}X^{\vee})$ whose every irreducible component has same dimension as $H$, and [6, Theorem 2.5] implies that $V\subset\mathbb{P}(T_{x}X^{\vee})$ is non-degenerate. Since ${\rm dim}\,H\geq\frac{1}{2}{\rm dim}\,\mathbb{P}(T_{x}X^{\vee})$, we have $H=V$, thus we obtain the conclusion. ∎

Lemma 2.6**.**

If $\rho_{X}=1$ and $H\subsetneq\mathbb{P}(T_{x}X^{\vee})$ is a linear space of positive dimension, then ${\rm dim}\,H\leq\frac{n-4}{2}$.

Proof.

Set $d:={\rm dim}\,H(\geq 1)$. Lemma 2.3 implies that $X$ is covered by linear spaces of dimension $d+1(\geq 2)$. By the proof of [7, Proposition 2.1], we see that the normal bundles of the linear spaces are trivial. On the other hand, if $d\geq\frac{n-3}{2}$ (i.e., $d+1\geq\frac{n-1}{2}$), then the normal bundles cannot be trivial by [11, Proposition 6.4]. Thus we obtain $d\leq\frac{n-4}{2}$. ∎

3. Higher order families of lines

In [14], we introduced higher order minimal families of rational curves and invariants $\overline{N}_{X}$ and $\underline{N}_{X}$ for Fano manifolds $X$. We introduce line versions of them as follows:

Definition 3.1**.**

We write $X\models H$ if $X$ is an embedded Fano manifold covered by lines and $H$ is a family of lines through a fixed general point. If $X\models H_{1}$ and $H_{1}$ is also a Fano manifold covered by lines in the projectivised tangent space, then we consider $H_{1}\models H_{2}$ a family of lines on $H_{1}$ through a fixed general point. In the same way, if there is a chain

[TABLE]

of length $m$, we call $H_{m}$ an $m$-th order family of lines. In addition, we denote by $S_{X}$ the greatest number $m$ such that there is an $m$-th order family of lines.

Example 3.2**.**

By Example 2.2, we have the following:

1. (1)

If $X$ is a linear space $\mathbb{P}^{n}$, then

[TABLE]

so $S_{X}=n$. 2. (2)

If $X$ is a quadric hypersurface $Q^{n}$, then

[TABLE]

so $S_{X}=\begin{cases}\frac{n}{2}&\text{ if }n\text{ is even}\\ \frac{n-1}{2}&\text{ if }n\text{ is odd}\end{cases}$. 3. (3)

If $X$ is the Grassmannian $G(2,\mathbb{C}^{m+2})$ (${\rm dim}\,X=2m$), then

[TABLE]

so $S_{X}=m$. 4. (4)

If $X$ is the symplectic Grassmannian $SG(2,\mathbb{C}^{m+3})$ (${\rm dim}\,X=2m+1$), then

[TABLE]

so $S_{X}=m$.

Proposition 3.3**.**

Any Fano manifold $X\subset\mathbb{P}^{N}$ is covered by linear spaces of dimension at least $S_{X}$.

Proof.

Set $m:=S_{X}$. By definition of $S_{X}$, there is a chain

[TABLE]

of length $m$. By applying Lemma 2.3, we inductively see that $H_{i}$ is covered by linear spaces of dimension $m-i$. ∎

Example 3.4**.**

There is a Fano manifold $X\subset\mathbb{P}^{N}$ which is covered by linear spaces of dimension greater than $S_{X}$. For example, let $X$ be a complete intersection of two quadric hypersurfaces in $\mathbb{P}^{7}$. Then $X\models H$ is a complete intersection of two quadric hypersurfaces in $\mathbb{P}^{4}$. Since $H$ contains a line, Lemma 2.3 yields that $X$ is covered by linear spaces of dimension $2$. However, $H$ is not covered by lines, so $S_{X}=1$.

Proposition 3.5**.**

Let $X\subset\mathbb{P}^{N}$ be a Fano $n$-fold.

1. (1)

$0\leq S_{X}\leq n$* holds.* 2. (2)

If $S_{X}=n$, then $X$ is a linear space $\mathbb{P}^{n}$. 3. (3)

If $n\geq 2$ and $S_{X}=n-1$, then $(X,\mathscr{O}_{\mathbb{P}^{N}}(1)|_{X})$ is isomorphic to either of the following:

• (i)

$(\mathbb{P}^{1}\times\mathbb{P}^{n-1},\mathscr{O}(d,1))$* for some $d\geq 1$;*

• (ii)

$(\mathbb{P}(\mathscr{O}_{\mathbb{P}^{1}}(d+1)\oplus{\mathscr{O}_{\mathbb{P}^{1}}(d)}^{n-1}),\mathscr{O}(1))$* for some $d\geq 1$, where $\mathscr{O}(1)$ is the tautological line bundle.*

Proof.

(1) and (2) follow from Proposition 3.3. We assume $S_{X}=n-1\geq 1$. Then, Proposition 3.3 and Theorem 1.3 imply that $X$ is a linear $\mathbb{P}^{n-1}$-bundle, so we get a vector bundle $\mathscr{E}$ on $\mathbb{P}^{1}$ such that $(X,\mathscr{O}_{\mathbb{P}^{N}}(1)|_{X})$ is isomorphic to $(\mathbb{P}(\mathscr{E}),\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1))$. We can write $\mathscr{E}\cong\bigoplus_{i=1}^{n}\mathscr{O}_{\mathbb{P}^{1}}(a_{i})$ for some positive integers $a_{1}\leq\cdots\leq a_{n}$. Since $X\cong\mathbb{P}(\mathscr{E})$ is a Fano manifold, we see that $\sum_{i=1}^{n}a_{i}\leq na_{1}+1$. Thus we obtain either (i) or (ii). ∎

Proposition 3.6** ((1) and (2) of Theorem 1.1).**

Let $X\subset\mathbb{P}^{N}$ be a Fano $n$-fold of Picard number $1$.

1. (1)

If $S_{X}>\frac{n}{2}$, then $X$ is a linear space $\mathbb{P}^{n}$. 2. (2)

If $S_{X}=\frac{n}{2}$, then $X$ is either a quadric hypersurface $Q^{2m}$ or the Grassmannian $G(2,\mathbb{C}^{m+2})$ $(n=2m\geq 4)$.

Proof.

The arguments immediately follow from Proposition 3.3 and Theorem 1.3. ∎

4. Main result

In this section, we prove (3) of Theorem 1.1:

Theorem 4.1**.**

Let $X\subset\mathbb{P}^{N}$ be a Fano manifold of Picard number $1$ and dimension $n=2m+1\geq 3$. Assume that Conjecture 1.2 is true and $S_{X}=m$. Then $X$ is isomorphic to one of the following:

• (a)

a quadric hypersurface $Q^{2m+1}$;

• (b)

the symplectic Grassmannian $SG(2,\mathbb{C}^{m+3})$;

• (c)

a cubic hypersurface in $\mathbb{P}^{4}$ $(m=1)$;

• (d)

a complete intersection of two quadric hypersurfaces in $\mathbb{P}^{5}$ $(m=1)$;

• (e)

a $3$-dimensional linear section of the Grassmannian $G(2,\mathbb{C}^{5})$ in $\mathbb{P}^{9}$ $(m=1)$.

Proof.

Since $S_{X}=m$, there is a chain

[TABLE]

of length $m$. Set $n_{i}:={\rm dim}\,H_{i}$. Since $X$ is not a linear space, Lemma 2.4(1) yields $n_{0}-n_{1}\geq 2$. Let $P_{i}$ be the projectivised tangent space of $H_{i-1}$ containing $H_{i}$. Note that $P_{i}$ is isomorphic to a projective space of dimension $n_{i-1}-1$.

Case 1**.**

$H_{i}$ is not a linear space in $P_{i}$ for any $1\leq i\leq m-1$.

In this case, $n_{i}-n_{i+1}\geq 2$ for any $1\leq i\leq m-1$, so we have:

[TABLE]

Thus, Lemma 2.4 implies that $X$ is one of the following:

• (i)

a quadric hypersurface $Q^{2m+1}$;

• (ii)

a cubic hypersurface in $\mathbb{P}^{2m+2}$;

• (iii)

a complete intersection of two quadric hypersurfaces in $\mathbb{P}^{2m+3}$;

• (iv)

a $3$-dimensional linear section of the Grassmannian $G(2,\mathbb{C}^{5})$ in $\mathbb{P}^{9}$ ($m=1$);

• (v)

a hyperplane section of the Grassmannian $G(2,\mathbb{C}^{5})$ in $\mathbb{P}^{9}$ ($m=2$).

According to the above inequality, if $n_{0}-n_{1}=3$ and $m\geq 2$, then $n_{1}-n_{2}=2$ must holds. However, in case (ii) (resp. (iii)), if $m\geq 2$, then $H_{1}$ is a complete intersection of a cubic hypersuface and a quadric hypersurface (resp. two quadric hypersufaces) in $\mathbb{P}^{2m}$, so we cannot take $H_{2}$ when $m=2$, and $n_{1}-n_{2}=4$ (resp. $3$) when $m\geq 3$. Thus, in case (ii) (resp. (iii)), we obtain $m=1$, so $X$ is (c) (resp. (d)). Notice that (v) is isomorphic to (b) $SG(2,\mathbb{C}^{5})$.

Case 2**.**

$H_{i}$ is a linear space in $P_{i}$ for some $1\leq i\leq m-1$.

In this case, $H_{i+j}(=P_{i+j})$ has dimension $n_{i}-j$ for any $j\geq 1$, and $H_{m}$ is a point, so $n_{i}=m-i$. By taking the minimum $i$, we may assume that $H_{i-1}$ is not a linear space in $P_{i-1}$.

We show $\rho_{H_{i-1}}\geq 2$. We assume by contradiction that $\rho_{H_{i-1}}=1$. Then Lemma 2.6 implies $\frac{n_{i-1}-4}{2}\geq n_{i}(=m-i)$, so

[TABLE]

a contradiction. Thus we obtain $\rho_{H_{i-1}}\geq 2$. In particular, $i\geq 2$ and $m\geq 3$. Moreover, since $H_{i-1}$ is embedded in a projective space $P_{i-1}$ of dimension $n_{i-2}-1$, we have $\frac{n_{i-2}}{2}\geq n_{i-1}(\geq 2+m-i)$ by Barth and Larsen’s theorem [2].

We know that none of the manifolds in Lemma 2.4 satisfies the condition of this case, so $n_{0}-n_{1}\geq 4$. Hence, if $i\geq 3$, then

[TABLE]

a contradiction. Thus we get $i=2$, and the inequalities $\frac{n_{0}}{2}\geq n_{1}\geq m$ yield $n_{1}=m$. By applying Lemma 2.5, we have that $H_{1}\subset P_{1}$ is non-degenerate. Furthermore, since $S_{H_{1}}=m-1=n_{1}-1$, Proposition 3.5 implies that $(H_{1},\mathscr{O}_{P_{1}}(1)|_{H_{1}})$ is isomorphic to either of the following:

• (i)

$(\mathbb{P}^{1}\times\mathbb{P}^{m-1},\mathscr{O}(d,1))$ for some $d\geq 1$;

• (ii)

$(\mathbb{P}(\mathscr{O}_{\mathbb{P}^{1}}(d+1)\oplus{\mathscr{O}_{\mathbb{P}^{1}}(d)}^{m-1}),\mathscr{O}(1))$ for some $d\geq 1$, where $\mathscr{O}(1)$ is the tautological line bundle.

Since $H_{1}$ is embedded in a projective space $P_{1}$ of dimension $2m$, in both cases, we have $d=1$ by the following Lemma 4.2. In case (i), the linear span of $H_{1}$ in $P_{1}$ has dimension $2m-1$, so $H_{1}\subset P_{1}$ is degenerate, that is a contradiction. Therefore, the case (i) does not occur. On the other hand, in case (ii), we obtain that $X$ is isomorphic to (b) $SG(2,\mathbb{C}^{m+3})$ by applying Conjecture 1.2. ∎

Lemma 4.2** (see [3, Lemma 3.1]).**

1. (1)

Let $S$ be $\mathbb{P}^{1}\times\mathbb{P}^{m-1}$ embedded in $\mathbb{P}^{dm+m-1}$ by $\mathscr{O}(d,1)$. If $d\geq 2$, then the secant variety of $S$ has dimension $2m+1$. 2. (2)

Let $S$ be $\mathbb{P}(\mathscr{O}_{\mathbb{P}^{1}}(d+1)\oplus{\mathscr{O}_{\mathbb{P}^{1}}}(d)^{m-1})$ embedded in $\mathbb{P}^{dm+m}$ by the tautological line bundle. If $d\geq 2$, then the secant variety of $S$ has dimension $2m+1$.