Verlinde bundles of families of hypersurfaces and their jumping lines

Abstract

Verlinde bundles are vector bundles $V_k$ arising as the direct image $\pi _*(\mathscr {L}^{\otimes k})$ of polarizations of a proper family of schemes $\pi :\mathfrak X \rightarrow S$. We study the splitting behavior of Verlinde bundles in the case where $\pi$ is the universal family $\mathfrak X \rightarrow \left|{\mathscr {O}(d)}\right|$ of hypersurfaces of degree d in $\left|{\mathscr {O}(d)}\right|$ and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles $V_{d+1}$ in the cases $n=2,3$.

1 Introduction

Let $\pi :\mathfrak X \rightarrow S$ be a proper family of schemes with a polarization $\mathscr {L}$. For $k\ge 1$, if the sheaf $\pi _*(\mathscr {L}^{\otimes k})$ is locally free, we call it the k-th Verlinde bundle of the family $\pi$.

For example (Iyer 2013), let $C\rightarrow T$ be a smooth projective family of curves of fixed genus. Consider the relative moduli space $\pi :{\text {SU}}(r)\rightarrow T$ of semistable vector bundles of rank r and trivial determinant. This family is equipped with a polarization $\varTheta$, the determinant bundle. The Verlinde bundles $\pi _*(\varTheta ^k)$ of this family are projectively flat (Hitchin 1990; Axelrod et al. 1991), and their rank is given by the Verlinde formula.

In this article, we study the example of the universal family $\pi :\mathfrak {X} \rightarrow \left|\mathscr {O}_{\mathbb {P}^n}(d)\right|$ of hypersurfaces of degree d in the complex projective space $\mathbb {P}^n$, with $n>1$. This family comes equipped with the polarization $\mathscr {L}$ given by the pullback of $\mathscr {O}(1)$ along the projection map $\mathfrak {X} \rightarrow \mathbb {P}^n$. For $k\ge 1$, the sheaf $\pi _{*}\mathscr {L}^{\otimes k}$ is locally free, as can be seen by considering the structure sequence of an arbitrary hypersurface of degree d in $\mathbb {P}^n$. For $k\ge 1$, we denote the k-th Verlinde bundle of the family $\pi$ by $V_k$.

To better understand $V_k$ we study its splitting type when restricted to lines in $\left|{\mathscr {O}(d)}\right|$.

Let $T\subseteq \left|{\mathscr {O}(d)}\right|$ be a line. On $T=\mathbb {P}^1$, we define the vector bundle $V_{k,T}:=V_k|_T$. The splitting type of $V_{k,T}$ is the unique non-increasing tuple $(b_1,\ldots , b_{r^{(k)}})$ of size $r^{(k)}:={\text {rk}}V_k$ such that $V_{k,T} \simeq \bigoplus _i \mathscr {O}(b_i)$.

Sequence (2.1) puts constraints on the $b_i$: they are all non-negative and they sum up to $d^{(k)}:=\deg (V_k)$. The set of such tuples $(b_i)$ can be ordered by defining the expression $(b'_i) \ge (b_i)$ to mean

$$\begin{aligned} \sum _{i=1}^s b'_i \ge \sum _{i=1}^s b_i \quad \text { for all }\quad s=1,\ldots , r. \end{aligned}$$

With this definition, smaller types are more general: the vector bundle $\mathscr {O} (b_i)$ on $\mathbb {P}^1$ specializes to $\mathscr {O} (b'_i)$ in the sense of Shatz (1976) if and only if $(b'_i) \ge (b_i).$

If $d^{(k)} \le r^{(k)}$, then the most generic possible type has thus the form $(1,\ldots ,1,0,\ldots ,0)$. We call this the generic splitting type. A computation shows that $d^{(k)} \le r^{(k)}$ if $k\le 2d$.

We have the following result on the cohomology class of the set of jumping lines

$$\begin{aligned} Z:=\left\{ T\in \mathbb G \mathrm r(1,\left|{\mathscr {O}(d)}\right|)\mid V_{d+1,T}\quad \text { has non-generic type}\right\} \end{aligned}$$

in the Grassmannian of lines in $\left|{\mathscr {O}(d)}\right|$:

Theorem 1

Let $n \le 3$, let Z be set of jumping lines of $V_{d+1}$, and let [Z] be the class of Z in the Chow ring ${\text {CH}}(\mathbb G \mathrm r(1,\left|{\mathscr {O}(d)}\right|))$. We have

$$\begin{aligned} \dim Z = n+1+\left( {\begin{array}{c}d-1+n\\ n\end{array}}\right) . \end{aligned}$$

Furthermore, let b range over the integers with the property $0\le b < \frac{\dim Z}{2}$ and define $a=\dim Z - b, a'=a+\frac{{{\mathrm{codim}}}Z-\dim Z}{2}$, $b'=b+\frac{{{\mathrm{codim}}}Z-\dim Z}{2}$.

• 1. If $\dim Z$ is odd or $n=2$, we have

  $$\begin{aligned}{}[Z] = \sum _{a,b} \left( {\left( {\begin{array}{c}a+1\\ n\end{array}}\right) }{\left( {\begin{array}{c}b+1\\ n\end{array}}\right) }-{\left( {\begin{array}{c}a+2\\ n\end{array}}\right) }{\left( {\begin{array}{c}b\\ n\end{array}}\right) }\right) \sigma _{a',b'}. \end{aligned}$$

  (1.1)

• 2. If $\dim Z$ is even and $n=3$, we have

  $$\begin{aligned}{}[Z]&= \sum _{a,b} \left( {\left( {\begin{array}{c}a+1\\ n\end{array}}\right) }{\left( {\begin{array}{c}b+1\\ n\end{array}}\right) }-{\left( {\begin{array}{c}a+2\\ n\end{array}}\right) }{\left( {\begin{array}{c}b\\ n\end{array}}\right) }\right) \sigma _{a',b'} \nonumber \\&\quad + \left( {\begin{array}{c}\frac{\dim Z}{2} + 2\\ n\end{array}}\right) \left( {\begin{array}{c}\frac{\dim Z}{2}\\ n\end{array}}\right) \sigma _{\frac{\dim Z}{2},\frac{\dim Z}{2}}. \end{aligned}$$

The computation is carried out by the method of undetermined coefficients, leading into various calculations in the Chow ring of the Grassmannian. The assumption $n\le 3$ is needed for a certain dimension estimation.

2 Attained splitting types

There exists a short exact sequence of vector bundles on $\left|{\mathscr {O}(d)}\right|$

$$\begin{aligned} 0\rightarrow \mathscr {O}(-1) \otimes H^0(\mathbb {P}^n, \mathscr {O}(k-d)) \xrightarrow {M} \mathscr {O} \otimes {H^0(\mathbb {P}^n, \mathscr {O}(k)) } \rightarrow {V_k} \rightarrow 0, \end{aligned}$$

(2.1)

as can be seen by taking the pushforward of a twist of the structure sequence of $\mathfrak X$ on $\mathbb {P}^n \times \left|{\mathscr {O}(d)}\right|$. The map M is given by multiplication by the section

$$\begin{aligned} \sum _I \alpha _I \otimes x^I \in H^0(\left|{\mathscr {O}(d)}\right|,\mathscr {O}(1)) \otimes H^0(\mathbb {P}^n,\mathscr {O}(d)). \end{aligned}$$

In particular, we have $r^{(k)} = \left( {\begin{array}{c}k+n\\ n\end{array}}\right) - \left( {\begin{array}{c}k+n-d\\ n\end{array}}\right)$ and $d^{(k)}=\left( {\begin{array}{c}k+n-d\\ n\end{array}}\right)$.

Lemma 1

Let $\mathscr {E}$ be a free $\mathscr {O}_{\mathbb {P}^1}$-module of finite rank, and let

$$\begin{aligned} 0 \rightarrow \mathscr {E}' \xrightarrow {\varphi } \mathscr {E} \xrightarrow {\psi } \mathscr {E}'' \rightarrow 0 \end{aligned}$$

be a short exact sequence of $\mathscr {O}_{\mathbb {P}^1}$-modules. Given a splitting $\mathscr {E}'' = \mathscr {E}''_1 \oplus \mathscr {O}$, we may construct a splitting $\mathscr {E} = \mathscr {E}_1 \oplus \mathscr {O}$ such that the image of $\varphi$ is contained in $\mathscr {E}_1$.

Proof

Define $\mathscr {E}_1 :=\ker (\mathrm {pr}_2\circ \psi )$, which is a locally free sheaf on $\mathbb {P}^1$. By comparing determinants in the short exact sequence $0 \rightarrow \mathscr {E}_1 \rightarrow \mathscr {E} \rightarrow \mathscr {O} \rightarrow 0$ we see that $\mathscr {E}_1$ is free, hence by an ${\text {Ext}}^1$ computation the sequence splits. The property ${\text {im}}(\varphi ) \subseteq \mathscr {E}_1$ follows from the definition. $\square$

Proposition 1

Let $f_1, f_2 \in \left|{\mathscr {O}(d)}\right|$ span the line $T \subseteq \left|{\mathscr {O}(d)}\right|$ and let p be the number of zero entries in the splitting type of $V_{k,T}$. We have

$$\begin{aligned} p = \dim H^0(\mathbb {P}^n,\mathscr {O}(k)) - \dim ({f_1 U + f_2 U}). \end{aligned}$$

Proof

Note that the map $M|_T$ sends a local section $\xi \otimes \theta$ to $s\xi \otimes f_1 \theta + t\xi \otimes f_2 \theta$. In particular, the image of $\mathscr {O}(-1)\otimes U$ is contained in $\mathscr {O} \otimes (f_1 U + f_2 U)$. It follows that $p \ge \dim H^0(\mathbb {P}^n,\mathscr {O}(k)) - \dim ({f_1 U + f_2 U})$.

To prove the other inequality, consider the induced sequence

$$\begin{aligned} 0 \rightarrow \mathscr {O}(-1)\otimes U \xrightarrow {M|_T} \mathscr {O}\otimes (f_1 U + f_2 U) \rightarrow \mathscr {E}'' \rightarrow 0 \end{aligned}$$

and assume for a contradiction that $\mathscr {E}'' \simeq \mathscr {E}_1''\oplus \mathscr {O}.$ By Lemma 1, we have a splitting $\mathscr {O}\otimes (f_1 U + f_2 U) \simeq \mathscr {E}_1 \oplus \mathscr {O}$ such that ${\text {im}}(M|_T) \subseteq \mathscr {E}_1$.

Consider the map ${\widetilde{M}}|_T :(\mathscr {O} \otimes U) \oplus (\mathscr {O} \otimes U) \rightarrow \mathscr {O} \otimes (f_1 U + f_2 U)$ defined by

$$\begin{aligned} {\widetilde{M}}|_T(a\otimes \theta _1,b \otimes \theta _2)=a\otimes f_1 \theta _1 + b \otimes f_2 \theta _2. \end{aligned}$$

We obtain the matrix description of ${\widetilde{M}}|_T$ from the matrix description of $M|_T$ as follows. If $M|_T$ is represented by the matrix A with coefficients $A_{i,j} = \lambda _{i,j} s + \mu _{i,j} t$, then ${\widetilde{M}}|_T$ is represented by a block matrix

$$\begin{aligned} B = \left( \begin{array}{c|c} A' &{} A'' \\ \end{array} \right) \end{aligned}$$

with $A'_{i,j} = \lambda _{i,j}$ and $A''_{i,j} = \mu _{i,j}$.

The property ${\text {im}}(M|_T)\subseteq \mathscr {E}_1$ implies that after some row operations, the matrix A has a zero row. By the construction of ${\widetilde{M}}|_T$, the same row operations lead to the matrix B having a zero row, but this is a contradiction, since the map ${\widetilde{M}}|_T$ is surjective. $\square$

Corollary 1

Let $T \subseteq \left|\mathscr {O} (d)\right|$ be a line spanned by the polynomials $f_1,f_2$. Assume that $d^{(k)} \le r^{(k)}$. Let $\theta$ range over a monomial basis of $H^0(\mathbb {P}^n, \mathscr {O}(k-d))$. The bundle $V_{k,T}$ has the generic splitting type if and only if $\langle f_1\theta ,f_2\theta \mid \theta \rangle$ is a linearly independent set in $H^0(\mathbb {P}^n,\mathscr {O}(k))$. $\square$

Corollary 2

Let $T \subseteq \left|\mathscr {O} (d)\right|$ be a line spanned by the polynomials $f_1,f_2$, and let $d^{(k)} \le r^{(k)}$. The bundle $V_{k,T}$ has not the generic type if and only if $\deg (\gcd (f_1,f_2)) \ge 2d-k$. In particular, if $d^{(k)} \le r^{(k)}$ but $k>2d$ then the generic type never occurs.

Proof

By Corollary 1, the bundle $V_{k,t}$ has non-generic type if and only if there exist linearly independent $g_1,g_2\in H^0(\mathbb {P}^n,\mathscr {O}(k-d))$ such that $g_1f_1+g_2f_2 = 0$. Let $h :=\gcd (f_1,f_2)$ and $d':=\deg h$.

If $d' \ge 2d-k$ then $\deg (f_i/h) \le k-d$ and we may take $g_1,g_2$ to be multiples of $f_1/h$ and $f_2/h$, respectively.

On the other hand, given such $g_1$ and $g_2$, we have $f_1\mid g_2 f_2$, which implies $f_1/h \mid g_2$, hence $d-d'\le k-d$. $\square$

Proposition 2

Let $k=d+1$. No types of $V_k$ other than $(1,\ldots ,1,0,\ldots ,0)$ and $(2,1,\ldots ,1, 0,\ldots ,0)$ occur.

Proof

Assume that the type of $V_k$ at some line $(f_1,f_2)$ is other than the two above. Then the type has at least two more zero entries than the general type. By Proposition 1, we have $\dim \langle f_1 \theta , f_2 \theta \mid \theta \rangle \le 2d^{(k)}-2$, so we find $g_1,g_2,g'_1,g'_2\in H^0(\mathbb {P}^n,\mathscr {O}(1))$ and two linearly independent equations

$$\begin{aligned} g_1f_1 + g_2f_2&= 0 \\ g'_1 f_1 + g'_2 f_2&= 0, \end{aligned}$$

with both sets $(g_1,g_2), (g'_1,g'_2)$ linearly independent. From the first equation it follows that $f_1 = g_2 h$ and $f_2 = -g_1 h$, for some common factor h. Applying this to the second equation, we find $g'_1 g_2 = g'_2 g_1$, hence $g'_1 = \alpha g_1$ and $g'_2 = \alpha g_2$ for some scalar $\alpha$, a contradiction. $\square$

Corollary 3

Let $k=d+1,$ let $T\subset \left|\mathscr {O} (d)\right|$ be a line spanned by $f_1,f_2$. The type $(2,1,\ldots ,1,0,\ldots ,0)$ occurs if and only if $\deg (\gcd (f_1,f_2) \ge d-1$. $\square$

3 The cohomology class of the set of jumping lines

Definition 1

Let $k \ge 1$ and $(b_i)$ be a splitting type for $V_k$. We define the set $Z_{(b_i)}$ of all points $t\in \mathbb G \mathrm r(1,\left|{\mathscr {O}(d)}\right|)$ such that $V_{k,t}$ has splitting type $(b_i)$. For the set of points t where $V_{k,t}$ has generic splitting type, we also write $Z_{\text {gen}}$, and define the set of jumping lines$Z:=\mathbb G \mathrm r(1,\left|{\mathscr {O}(d)}\right|) {\setminus } Z_{gen}$.

Now let $k=d+1$. By Corollary 3, Z is the subvariety given as the image of the finite, generically injective multiplication map

$$\begin{aligned} \varphi :\mathbb G \mathrm r(1,\left|{\mathscr {O}(1)}\right|) \times \left|{\mathscr {O}(d-1)}\right| \rightarrow \mathbb G \mathrm r(1,\left|{\mathscr {O}(d)}\right|) \end{aligned}$$

sending the tuple $((sg_1+tg_2)_{(s:t)\in \mathbb {P}^1},h)$ to the line $(shg_1+thg_2)_{(s:t)\in \mathbb {P}^1}$.

To perform calculations in the Chow ring A of $\mathbb G \mathrm r(1,\left|{\mathscr {O}(d)}\right|)$, we follow the conventions found in Eisenbud and Harris (2016). We assume ${\text {char}}(k) = 0$ for simplicity. Let $N:=\dim H^0(\mathscr {O}(d)) = \left( {\begin{array}{c}n+d\\ n\end{array}}\right)$. For $N-2\ge a\ge b$, we have the Schubert cycle

$$\begin{aligned} \varSigma _{a,b}:=\{T \in \mathbb G \mathrm r(1,\left|{\mathscr {O}(d)}\right|) : T \cap H \ne \varnothing , T \subseteq H'\}, \end{aligned}$$

where $(H\subset H')$ is a general flag of linear subspaces of dimension $N-a-2$ resp. $N-b-1$ in the projective space $\left|{\mathscr {O}(d)}\right|$. The ring A is generated by the Schubert classes $\sigma _{a,b}$ of the cycles $\varSigma _{a,b}$. The class $\varSigma _{a,b}$ has codimension $a+b$, and we use the convention $\sigma _{a}:=\sigma _{a,0}$.

Proof

(of Theorem 1) We have $\dim Z = n+1+\left( {\begin{array}{c}d-1+n\\ n\end{array}}\right)$ since Z is the image of the generically injective map $\varphi$.

Let $Q \subset \left|{\mathscr {O}(d)}\right|$ be the image of the multiplication map

$$\begin{aligned} f:\left|{\mathscr {O}(1)}\right| \times \left|{\mathscr {O}(d-1)}\right| \rightarrow \left|{\mathscr {O}(d)}\right|. \end{aligned}$$

The map f is birational on its image, since a general point of Q has the form gh with h irreducible. The Chow group $A^{{{\mathrm{codim}}}Z}$ is generated by the classes $\sigma _{a',b'}$ with $N-2\ge a'\ge b' \ge \left\lfloor \frac{{{\mathrm{codim}}}Z}{2}\right\rfloor$ and $a'+b'={{\mathrm{codim}}}Z$, while the complementary group $A^{\dim Z}$ is generated by the classes $\sigma _{\dim Z-b,b}$ with $b\in {0,\ldots ,\left\lfloor \frac{\dim Z}{2}\right\rfloor }$. Write

$$\begin{aligned}{}[Z] = \sum _{a',b'} \alpha _{a',b'} \sigma _{a',b'}. \end{aligned}$$

We have $\sigma _{a',b'} \sigma _{a,b} = 1$ if $b'-b = \left\lfloor \frac{{{\mathrm{codim}}}Z}{2}\right\rfloor$ and 0 else. Hence, multiplying the above equation with the complementary classes $\sigma _{a,b}$ and taking degrees gives

$$\begin{aligned} \alpha _{a',b'} = \deg ([Z]\cdot \sigma _{a,b}). \end{aligned}$$

Using Giambelli’s formula $\sigma _{a,b}=\sigma _{a}\sigma _{b} - \sigma _{a+1}\sigma _{b-1}$ (Eisenbud and Harris 2016, Prop. 4.16), we reduce to computing $\deg ([Z] \cdot \sigma _{a}\sigma _{b})$ for $0\le b\le \left\lfloor \frac{\dim Z}{2}\right\rfloor$. By Kleiman transversality, we have

$$\begin{aligned} \deg ([Z] \cdot \sigma _{a}\sigma _{b}) = \left|\{T\in Z : T \cap H \ne \varnothing , T \cap H' \ne \varnothing \}\right|, \end{aligned}$$

where H and $H'$ are general linear subspaces of $\left|{\mathscr {O}(d)}\right|$ of dimension $N-a-2$ and $N-b-2$, respectively.

To a point $p = g_p h_p \in Q$ with $g_p \in \left|{\mathscr {O}(1)}\right|$ and $h_p \in \left|{\mathscr {O}(d-1)}\right|$, associate a closed reduced subscheme $\varLambda _p\subset Q$ containing p as follows. If $h_p$ is irreducible, let $\varLambda _p$ be the image of the linear embedding $\left|{\mathscr {O}(1)}\right|\times \{h_p\} \rightarrow \left|{\mathscr {O}(d)}\right|$ given by $g \mapsto g h_p$.

If $h_p$ is reducible, define the subscheme $\varLambda _p$ as the union $\bigcup _h {\text {im}}(\left|{\mathscr {O}(1)}\right| \times \{h\}\rightarrow \left|{\mathscr {O}(d)}\right|)$, where h ranges over the (up to multiplication by units) finitely many divisors of p of degree $d-1$.

Note that for all points p, the spaces ${\text {im}}(\left|{\mathscr {O}(1)}\right| \times \{h\}\rightarrow \left|{\mathscr {O}(d)}\right|)$ meet exactly at p.

By the definition of Z, all lines $T\in Z$ lie in Q. Furthermore, if T meets the point p, then $T\subseteq \varLambda _p$. For $H\subseteq \left|{\mathscr {O}(d)}\right|$ a linear subspace of dimension $N-a-2$, define $Q':=H\cap Q$. For general H, the subscheme $Q'$ is a smooth subvariety of dimension $b-n+1$ such that for a general point $p=gh$ of $Q'$ with $h\in \left|{\mathscr {O}(d)}\right|$, the polynomial h is irreducible.

Next, we consider the case $n=2$ or $\dim Z$ odd.

Claim

For genereal H, for each point $p\in Q'$ we have $\varLambda _p \cap H =\{p\}$.

Proof

(of Claim) Let $\mathscr {H}$ denote the Grassmannian ${\text {Gr}}(\dim H+1, N)$. Define the closed subset $X\subseteq Q\times \mathscr {H}$ by

$$\begin{aligned} X:=\{(p,H):\dim (H\cap \varLambda _p)\ge 1\}. \end{aligned}$$

The fibers of the induced map $X\rightarrow \mathscr {H}$ have dimension at least one. Hence, to prove that the desired condition on H is an open condition, it suffices to prove $\dim (X) \le \dim (\mathscr {H})$.

The fiber of the map $X\rightarrow Q$ over a point p consists of the union of finitely many closed subsets of the form $X'_p = \{H\in \mathscr {H} : \dim (H\cap \varLambda '_p)\ge 1\}$, where $\varLambda '_p\simeq \mathbb {P}^n\subseteq \left|{\mathscr {O}(d)}\right|$ is one of the components of $\varLambda _p$. The space $X'_p$ is a Schubert cycle

$$\begin{aligned} \varSigma _{\dim Q - b,\dim Q - b} = \{H\in {\text {Gr}}(\dim H+1, N) : \dim (H \cap H_{n+1}) \ge 2\}, \end{aligned}$$

with $H_{n+1}$ an $(n+1)$-dimensional subspace of $H^0(\mathscr {O}(d))$. The codimension of the cycle is $2(\dim Q - b)$, hence also ${{\mathrm{codim}}}(X_p) = 2(\dim Q -b)$. Finally, we have $\dim (\mathscr {H})-\dim (X) = {{\mathrm{codim}}}(X_p) - \dim (Q) = \dim Q - 2b$.

If $\dim Z$ is odd, then $\dim Q - 2b \ge \dim Q - \dim Z + 1 = 3-n\ge 0$. If $n=2$, we instead estimate $\dim Q - 2b \ge \dim Q - \dim Z = 2-n\ge 0$. $\square$

Next, let

$$\begin{aligned} \varLambda :=\bigcup _{p\in Q'} \varLambda _p = f(\left|{\mathscr {O}(1)}\right|\times \mathrm {pr}_2 f^{-1}(Q')) \end{aligned}$$

and

$$\begin{aligned} \varLambda '':=\left|{\mathscr {O}(1)}\right|\times \mathrm {pr}_2 f^{-1}(Q'). \end{aligned}$$

By the choice of H, the map $f^{-1}(Q')\rightarrow Q'$ is birational and the map $f^{-1}(Q')\rightarrow \mathrm {pr}_2f^{-1}(Q')$ is even bijective. It follows that $\varLambda ''$ and hence $\varLambda$ have dimension $b+1$.

The intersection of $\varLambda$ with a general linear subspace $H'$ of dimension $N-b-2$ is a finite set of points. For each point $p\in Q'$, the linear subspace $H'$ intersects each component $\varLambda '_p$ of $\varLambda _p$ in at most one point. For each point $p'\in H'\cap \varLambda$ there exists a unique p such that $p'\in \varLambda _p$.

The only line $T\in Z$ meeting both p and $H'$ is the one through p and $p'$. If the intersection $H'\cap \varLambda _p$ is empty, then there will be no line meeting p and $H'$. Hence, $\deg ([Z]\cdot \sigma _{a}\sigma _{b})$ is the number of intersection points of $\varLambda$ with a general $H'$.

Finally, the pre-image $f^{-1}(Q') = f^{-1}(H)$ is smooth for a general H by Bertini’s Theorem. If $\zeta$ is the class of a hyperplane section of $\left|{\mathscr {O}(d)}\right|$ we have $f^*(\zeta ) = \alpha + \beta$, where $\alpha$ and $\beta$ are classes of hyperplane sections of $\left|{\mathscr {O}(1)}\right|$ and $\left|{\mathscr {O}(d)}\right|$, respectively. Since $\mathrm {pr}_2$ and f have degree one, we compute

$$\begin{aligned}{}[\varLambda '']&= [\mathrm {pr}_2^{-1}\mathrm {pr}_2 f^{-1}(H)] = \mathrm {pr}_2^*\mathrm {pr}_{2,*}f^*[H] = {\left( {\begin{array}{c}{{\mathrm{codim}}}H\\ n\end{array}}\right) }\beta ^{{{\mathrm{codim}}}H -n}. \end{aligned}$$

Hence, by the push–pull formula:

$$\begin{aligned} \deg ([\varLambda ]\cdot H')&= \deg ([\varLambda '']\cdot (\alpha +\beta )^{{{\mathrm{codim}}}H'}) \\&= \left( {\begin{array}{c}{{\mathrm{codim}}}H\\ n\end{array}}\right) \left( {\begin{array}{c}{{\mathrm{codim}}}H'\\ n\end{array}}\right) =\left( {\begin{array}{c}a+1\\ n\end{array}}\right) \left( {\begin{array}{c}b+1\\ n\end{array}}\right) . \end{aligned}$$

We then use Giambelli’s formula to obtain Eq. 1.1.

In case $n=3$ and $\dim Z$ even, we show that for $b=\dim Z/2$ we have $\deg ([Z]\cdot \sigma _{b,b})=0$. In this case, the hyperplanes H and $H'$ have the same dimension $N-b-2$.

For $p\in Q$, the set $\varLambda _p$ is defined as before.

Claim

for general H of dimension $N-b-2$, we have $\dim (\varLambda _p\cap H)=1$.

Proof

(of Claim) Define as before the closed subset $X\subseteq Q\times \mathscr {H}$ by

$$\begin{aligned} X:=\{(p,H):\dim (H\cap \varLambda _p)\ge 1\}. \end{aligned}$$

The generic fiber of the projection map $\varphi :X\rightarrow \mathscr {H}$ is one-dimensional, hence we have $\dim \varphi (X) = \dim (X)-1 = \dim \mathscr {H}$. The last equation holds with $n=3$ and $2b=\dim Z$. Hence for all $H\in \mathscr {H}$ we have $\dim (\varLambda _p \cap H)\ge 1$.

On the other hand, the equality $\dim (\varLambda _p \cap H)=1$ is attained by some, and hence by a general, H. Indeed, Define the closed subset $X\subseteq Q\times \mathscr {H}$ by

$$\begin{aligned} X:=\{(p,H):\dim (H\cap \varLambda _p)\ge 1\}. \end{aligned}$$

By a similar argument as before, one needs to show that $\dim (\mathscr {H})-\dim (X) + 1\ge 0$. The fiber $X_p$ is a Schubert cycle of codimension $3(\dim Q-b+1)$. Lastly, a computation shows $\dim (\mathscr {H})-\dim ({\widetilde{X}})+1={{\mathrm{codim}}}({\widetilde{X}}_p)-\dim (Q)+1=\frac{1}{2}(2\dim Q + 18 - 5n)\ge 0$. $\square$

Now, define $\varLambda ''$ as above. We have $\dim \varLambda '' = \dim \left|{\mathscr {O}(1)}\right| + \dim \mathrm {pr}_2 f^{-1}(Q') = b$. Since f is generically of degree one, we still have $\dim \varLambda '' = \varLambda$, hence $\dim \varLambda + \dim H' = N-2 < \dim \left|{\mathscr {O}(d)}\right|$. It follows that a generic $H'$ does not meet any of the lines $T\subset Z$, hence $\sigma _{b}\sigma _{b}\cdot [Z] = 0$. $\square$