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. 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. 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.

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 \(\deg (\gcd (f_1,f_2) \ge d-1\). \(\square \)

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 \(\sigma _{b}\sigma _{b}\cdot [Z] = 0\). \(\square \)