1 Introduction↩︎

Let \(X\) be a smooth projective complex variety of dimension \(n\), and \(L\) be a line bundle on \(X\). Denote by \(X^{[k+1]}\) the Hilbert scheme of \(k+1\) points on \(X\), and consider the tautological bundle \(E_{k+1,L}\) on \(X^{[k+1]}\) associated to \(L\) (see Section 2 for the definitions and notations). The cohomology groups \(H^i(X^{[k+1]}, S^\ell E_{k+1,L})\) of symmetric powers of the tautological bundle \(E_{k+1,L}\) has attracted considerable attention (see e.g., [1]–[5]). This has been studied mainly for the case where \(n\) and \(\ell\) are small and \(k\) is arbitrary. In this paper, we deal with the case where \(n\) and \(\ell\) are arbitrary and \(k=1\) or \(2\) under the assumption that \(L\) is sufficiently positive (see Remark 5 for the discussion on the required positivity condition for \(L\)). Our result may provide a complement to the existing literature.

This theorem is a higher dimensional generalization of [3], which deals with the case where \(X\) is a curve. Since the cohomology groups \(H^i(X^{[2]}, \mathcal{O}_{X^{[2]}})\) can be explicitly determined (Lemma 3), the problem is reduced to computing the Hilbert function \(H_{\Sigma_k}(\ell)=h^0(\Sigma_k, \mathcal{O}_{\Sigma_k}(\ell))\) of the \(k\)-th secant variety \(\Sigma_k\) of \(X \subseteq \mathbb{P} H^0(X, L) = \mathbb{P}^r\) for \(k=1\) and \(2\), which is defined as \[\Sigma_k= \Sigma_k(X, L) := \overline{\bigcup_{x_1, \ldots, x_{k+1} \in X} \langle x_1, \ldots, x_{k+1} \rangle} \subseteq \mathbb{P}^r.\] Here as \(L\) is sufficiently positive, \(L\) is very ample giving an embedding of \(X\) into \(\mathbb{P}^r\). There has been a great deal of work on secant varieties in various flavors (see e.g., [3], [6]–[12]). Recently, many people are interested in the connection to the tensor decomposition problem and computational complexity theory (see [13], [14]). In the main cases of this paper, namely, \(k=1\) or \(2\), it is known that \(H^i(\Sigma_k, \mathcal{O}_{\Sigma_k}(\ell)) = 0\) for \(i>0\) and \(\ell > 0\) (see Theorem 4). Thus \(h^0(\Sigma_k, \mathcal{O}_{\Sigma_k}(\ell))= \chi(\mathcal{O}_{\Sigma_k}(\ell))\) for \(\ell \geq 1\). In other words, the Hilbert function \(H_{\Sigma_k}(\ell)\) coincides with the Hilbert polynomial \(P_{\Sigma_k}(\ell):=\chi(\mathcal{O}_{\Sigma_k}(\ell))\). In this paper, we express \(P_{\Sigma_k}(\ell)\) in terms of \(\Gamma:=\sum_{i=1}^n (-1)^n h^p(X, \mathcal{O}_X)\) and the following functions \[l(a):=\chi(L^a),~s_1(a,b):=\chi(S^a \Omega_X \otimes L^b),~s_2(a,b,c):=\chi(S^a \Omega_X \otimes S^b \Omega_X \otimes L^c)\] depending only on the cotangent bundle \(\Omega_X\) and the embedding line bundle \(L\).

This theorem is a higher dimensional generalization of [3], which deals with the case where \(X\) is a curve. As in the theorem, assume that \(k=1\) or \(2\) and \(L\) is sufficiently positive. Let \(B^{k+1}=B^{k+1}(L):=\mathbb{P}(E_{k+1,L})\). Note that \(H^i(X^{[k+1]}, S^{\ell} E_{k+1,L}) = H^i(B^{k+1}, \mathcal{O}_{B^{k+1}}(\ell))\) for \(i \geq 0\) and \(\ell \geq 1\). The complete linear system \(\lvert \mathcal{O}_{B^{k+1}}(1) \rvert\) induces a birational morphism \(\alpha_{k+1} \colon B^{k+1} \to \Sigma_k\), which is a resolution of singularities. We review basic facts on secant varieties in Section 2. If one can determine the higher direct images \(R^i \alpha_{k+1,*} \mathcal{O}_{B^{k+1}}\) explicitly, then Theorem 1 follows from the Leray spectral sequence for \(\alpha_{k+1}\) together with known results for the cohomology of secant varieties. This is precisely what we do in Section 3. We also briefly discuss in Remark 7 how to extend Theorem 1 when \(X\) is a surface and \(k\) is arbitrary. The idea of computing the cohomology groups of symmetric powers of the tautological bundles via secant varieties was already explored in [3] in the case where \(X\) is a curve. In that case, since \(\deg P_{\Sigma_k}(\ell)=\dim \Sigma_k = 2k+1\), the result is obtained by applying Lagrange interpolation after computing \(P_{\Sigma_k}(\ell)\) for \(-k \leq \ell \leq k+1\). These values can be derived rather easily from the descriptions of \(R^i \alpha_{k+1,*} \mathcal{O}_{B^{k+1}}\). However, in the higher dimensional cases, we need a new strategy (we instead compute \(P_{\Sigma_k}(km+k-1)\) for all \(m \geq 0\)), and the arguments become substantially more involved. See Section 4 for the details. Finally, we note that the degrees of \(\Sigma_1\) and \(\Sigma_2\) were known before ([15] and [8], respectively). Our result may be regarded as a far-reaching generalization of those work.

2 Preliminaries↩︎

We begin with collecting necessary facts on Hilbert schemes of points and secant varieties. We closely follow the notations in [7], and we refer to that paper and the references therein for further details. Throughout the paper, \(X\) is a smooth projective complex variety of dimension \(n\), and \(L\) is a sufficiently positive line bundle on \(X\) giving an embedding \(X \subseteq \mathbb{P} H^0(X, L) = \mathbb{P}^r\).

2.1 Hilbert schemes of points↩︎

The Hilbert scheme \(X^{[k]}:= \{ \xi \subseteq X \mid \dim \xi=0, \operatorname{length}(\xi)=k\}\) of \(k\) points on \(X\) is smooth if and only if \(n \leq 2\) or \(k \leq 3\) (see [16]). In this paper, we always assume that \(k \leq 3\) so that \(X^{[k]}\) is a smooth projective complex variety of dimension \(kn\). For a zero-dimensional subscheme \(\xi \subseteq X\) of length \(k\), we denote by \([\xi]\) the point of \(X^{[k]}\) corresponding to \(\xi\). The universal family \[\mathcal{Z}_{k}:=\{ (x, [\xi]) \in X \times X^{[k]} \mid x \in \xi\} \subseteq X \times X^{[k]}\] over \(X^{[k]}\) has rational singularities (see [7]). Let \(\textrm{pr}_1 \colon \mathcal{Z}_{k} \to X\) and \(\textrm{pr}_2 \colon \mathcal{Z}_{k} \to X^{[k]}\) be the projection maps, and \(E_{k,L}:=\textrm{pr}_{2,*} \textrm{pr}_1^* L\) be the tautological bundle associated to \(L\), which is a rank \(k\) vector bundle on \(X^{[k]}\) since \(\textrm{pr}_2\) is a finite flat morphism of degree \(k\). Note that \(H^0(X^{[k]}, E_{k,L}) = H^0(X, L)\) and the fiber of \(E_{k,L}\) over \([\xi] \in X^{[k]}\) can be naturally identified with \(H^0(\xi, L|_{\xi})\). In particular, \(E_{k,L}\) is globally generated whenever \(L\) is \((k-1)\)-very ample, i.e., \(H^0(X, L) \to H^0(\xi, L|_{\xi})\) is surjective for every \([\xi] \in X^{[k]}\).

Next, we consider the nested Hilbert scheme \[X^{[k-1,k]}:=\{ ([\eta], [\xi]) \in X^{[k-1]} \times X^{[k]} \mid \eta \subseteq \xi \} \subseteq X^{[k-1]} \times X^{[k]}\] with projection maps \(\tau \colon X^{[k-1, k]} \to X^{[k-1]}\) and \(\rho \colon X^{[k-1,k]} \to X^{[k]}\). There is a residual morphism \(\mathrm{res}\colon X^{[k-1,k]} \to X\) sending \(([\eta], [\xi])\) to \((\mathscr{I}_{\xi}: \mathscr{I}_{\eta})\). When \(n \leq 2\) or \(k \leq 3\), it is also known that \(X^{[k-1,k]}\) is a smooth projective complex variety of dimension \(kn\) (see [16]). The morhpism \(\rho \colon X^{[k-1,k]} \to X^{[k]}\) is generically finite and factors through the universal family \(\mathcal{Z}_k\). The morphism \(X^{[k-1,k]} \to \mathcal{Z}_k\) is a resolution of singularities. On the other hand, \(X^{[k-1,k]}\) can be obtained by the blow-up \(\textrm{bl}\colon X^{[k-1, k]} \to X \times X^{[k-1]}\) along the universal family \(\mathcal{Z}_{k-1}\) with exceptional divisor \(F_{k-1}\). Then \(\mathrm{res}= \textrm{pr}_1 \circ \textrm{bl}\) and \(\tau = \textrm{pr}_2 \circ \textrm{bl}\), where \(\textrm{pr}_1 \colon X \times X^{[k-1]} \to X\) and \(\textrm{pr}_2 \colon X \times X^{[k-1]} \to X^{[k-1]}\) are projection maps. Notice that \(\mathcal{Z}_2 = X^{[1,2]}\) is obtained by the blow-up of \(X \times X\) along the diagonal \(\Delta \subseteq X \times X\). We have a short exact sequence \[0 \longrightarrow \mathrm{res}^* L(-F_{k-1}) \longrightarrow \rho^* E_{k,L} \longrightarrow \tau^* E_{k-1,L} \longrightarrow 0.\]

Now, we consider the Hilbert–Chow morphism \(h_k \colon X^{[k]} \to X^{(k)}\). The \(k\)-th symmetric product \(X^{(k)}\) is obtained by the quotient of the ordinary \(k\)-th product \(X^k\) by the symmetric group action \(\mathfrak{S}_k\) permuting the components. The line bundle \(L^{\boxtimes k}\) on \(X^k\) descends to a line bundle \(S_{k,L}\) on \(X^{(k)}\). We denote by \(T_{k,L}:=h_k^* S_{k,L}\). Then there exists a divisor \(\delta_k\) on \(X^{[k]}\) such that \(N_{k,L}:=\det E_{k,L} = T_{k,L}(-\delta_k)\). Notice that \(2\delta_k\) is the exceptional divisor of \(h_k\) when \(n \geq 2\). We set \(A_{k,L}:=T_{k,L}(-2\delta_k)\). When \(k=1\), we have \(\delta_1=0\) and \(E_{1,L}=T_{1,L}=N_{1,L}=A_{1,L}=L\). We refer to [7] for basic properties of these line bundles on \(X^{[k]}\). Here we recall that \(\rho^* T_{k,L} = \tau^* T_{k-1,L} \otimes \mathrm{res}^* L\) and \(\rho^* \delta_k = \tau^* \delta_{k-1} + F_{k-1}\). We can also compute the canonical line bundles \(\omega_{X^{[k]}} = T_{k, \omega_X}((n-2)\delta_k)\) and \(\omega_{X^{[k-1,k]}} = (\tau^* \omega_{X^{[k-1]}} \otimes \mathrm{res}^*\omega_X)((n-1)F_{k-1})\) (see [7]). Notice that \(F_{k-1}\) is the ramification divisor of \(\rho\) and \(\omega_{X^{[k-1,k]}} = \rho^* \omega_{X^{[k]}}(F_{k-1})\).

Finally, we show the following lemma, which is certainly known to experts.

::: {#lem:H^i(O_X[2]) .lem} Lemma 3. We have \[H^i(X^{[2]}, \mathcal{O}_{X^{[2]}}) = \Bigg( \bigoplus_{\substack{p+q=i \\ p<q}} H^p(X, \mathcal{O}_X) \otimes H^q(X, \mathcal{O}_X) \Bigg) \oplus \begin{cases} S^2 H^{i/2}(X, \mathcal{O}_X) & \text{if i is even and i/2 is even} \\ \wedge^2 H^{i/2}(X, \mathcal{O}_X) & \text{if i is even and i/2 is odd} \\ 0 & \text{i is odd}. \end{cases}\] :::

2.2 Secant varieties↩︎

We keep assuming \(k=2\) or \(3\) so that \(X^{[k]}\) and \(X^{[k-1,k]}\) are smooth. Most of the results still holds when \(n \leq 2\) or \(k \leq 3\). Let \(B^k=B^k(L):=\mathbb{P}(E_{k+1,L})\) with canonical projection \(\pi_k \colon B^k \to X^{[k]}\), and \(H_k\) be the tautological divisor on \(B^k\) so that \(\mathcal{O}_{B^k}(H_k) = \mathcal{O}_{B^k}(1)\). As \(E_{k,L}\) is globally generated, the tautological line bundle \(\mathcal{O}_{B^k}(1)\) is base point free. Note that \(H^0(B^k, \mathcal{O}_{B^k}(1)) = H^0(X^{[k]}, E_{k,L})=H^0(X, L)\). Thus the complete linear system \(\lvert \mathcal{O}_{B^k}(1) \rvert\) gives a morhpism \(\alpha_k \colon B^k \to \mathbb{P} H^0(X, L) = \mathbb{P}^r\) whose image is the \(k\)-secant variety \[\sigma_k=\sigma_k(X, L):=\overline{\bigcup_{[\xi] \in X^{[k]}} \langle \xi \rangle} \subseteq \mathbb{P}^r,\] which is nothing but the \((k-1)\)-th secant variety \(\Sigma_{k-1}=\Sigma_{k-1}(X, L)\). Two different conventions for secant varieties are both widely used in the literature. From now on, following [7], we use the notation of \(k\)-secant variety \(\sigma_k\) simply because it is more convenient for maintaining consistent indexing. Note that \(\alpha_k \colon B^k \to \sigma_k\) is a birational morphism hence a resolution of singularities and \(\sigma_k\) is a projective variety with \(\dim \sigma_k = kn+k-1\). We have \(\operatorname{Sing}(\sigma_k)=\sigma_{k-1}\) (see [7]). Here, recall some of the main results of [7] (see also [9] and [12] for the case of \(k=2\)).

Next, let \(B^{k-1,k}=B^{k-1,k}(L):=\mathbb{P}(\tau^* E_{k-1,L})\) with canonical projection \(\pi_{k-1,k} \colon B^{k-1,k} \to X^{[k]}\). We have a commutative diagram \[\xymatrix{ B^{k-1,k} \ar[r]^-{\widetilde{\tau}} \ar[d]_-{\pi_{k-1,k}} & B^{k-1} \ar[d]^-{\pi_{k-1}} \\ X^{[k-1,k]} \ar[r]_{\tau} & X^{[k-1]}. }\] On the other hand, from the surjection \(\rho^* E_{k,L} \to \tau^* E_{k-1,L}\), we get a morphism \(\alpha_{k-1,k} \colon B^{k-1,k} \to B^k\) whose image is denoted by \(Z_{k-1}^k\). Note that \(\alpha_{k-1,k} \colon B^{k-1,k} \to Z_{k-1}^k\) is a birational morphism hence a resolution of singularities and \(Z_{k-1}^k:=\alpha_k^{-1}(\sigma_{k-1})\) is the exceptional divisor of \(\alpha_k\). By [7], \(\mathcal{O}_{B^3}(-Z_{k-1}^k) = \mathcal{O}_{B^k}(-k) \otimes \pi_k^* A_{k,L}\). There is a commutative diagram \[\xymatrix{ B^{k-1,k} \ar[r]^-{\alpha_{k-1,k}} \ar[d]_-{\pi_{k-1,k}} & Z_{k-1}^k \ar[d]^-{\pi_k} \\ X^{[k-1,k]} \ar[r]_{\rho} & X^{[k]}. }\] Note that \(\alpha_k \circ \alpha_{k-1,k} = \alpha_{k-1} \circ \widetilde{\tau}\) and \(\alpha_k(\alpha_{k-1,k}(B^{k-1,k}))=\alpha_k(Z_{k-1}^k)=\sigma_{k-1}\). When \(k=2\), we have \(\mathcal{Z}_2 = X^{[1,2]} = B^{1,2} = Z_1^2\), which is the universal family over \(X^{[2]}\) via \(\pi_2\), and the restricted morphism \(\alpha_2 \colon Z_1^2 \to X\) coincides with the residual morhpism \(\mathrm{res}\colon X^{[1,2]} \to X\). For \(k=3\), consider the divisor \(\widetilde{\tau}^* Z_1^2\) on \(B^{2,3}\). Let \(Z_1^3:=\alpha_{2,3}(\widetilde{\tau}^*Z_1^2)\). Then \(\alpha_3(Z_1^3)=X\) and \(Z_1^3 = \alpha_3^{-1}(X)\). Note that \(Z_1^3 = \mathcal{Z}_3\) is the universal family over \(X^{[3]}\) via \(\pi_3\) and the restricted morphism \(\alpha_3 \colon Z_1^3 \to X\) coincides with the the projection map \(\textrm{pr}_1 \colon \mathcal{Z}_3 \to X\). Since \(Z_2^3\) is a prime divisor on a smooth variety \(B^3\), it is Gorenstein. We have \(\omega_{B^{2,3}}(\widetilde{\tau}^* Z_1^2) = \alpha_{2,3}^* \omega_{Z_2^3}\). Observe that \(\alpha_{2,3} \colon B^{2,3} \setminus \widetilde{\tau}^* Z_1^2 \to Z_2^3 \setminus Z_1^3\) is an isomorphism and \(\alpha_{2,3} \colon \widetilde{\tau}^* Z_1^2 \to Z_1^3\) is a generically 2-to-1 map. It is easy to show that \(\alpha_{2,3,*} \mathcal{O}_{B^{2,3}}(-\widetilde{\tau}^* Z_1^2) \rightarrow \mathcal{O}_{Z_2^3}\) is injective and the subscheme of \(Z_2^3\) defined by \(\alpha_{2,3,*} \mathcal{O}_{B^{2,3}}(-\widetilde{\tau}^* Z_1^2)\) is generically reduced. But \(\alpha_{2,3,*} \mathcal{O}_{B^{2,3}}(-\widetilde{\tau}^* Z_1^2)\) and \(\mathcal{O}_{Z_2^3}\) are Cohen–Macaulay, so \(\alpha_{2,3,*} \mathcal{O}_{B^{2,3}}(-\widetilde{\tau}^* Z_1^2) = \mathscr{I}_{Z_1^3/Z_2^3}\) since the cosupport of \(\alpha_{2,3,*} \mathcal{O}_{B^{2,3}}(-\widetilde{\tau}^* Z_1^2)\) is \(Z_1^3\). Thus we have a short exact sequence \[0 \longrightarrow \alpha_{2,3,*} \mathcal{O}_{B^{2,3}}(-\widetilde{\tau}^* Z_1^2) \longrightarrow \mathcal{O}_{Z_2^3} \longrightarrow \mathcal{O}_{Z_1^3} \longrightarrow 0.\] Using Kawamata–Viehweg vanishing theorem, one can easily check \(R^i \alpha_{2,3,*} \mathcal{O}_{B^{2,3}} = R^i \alpha_{2,3,*} \mathcal{O}_{B^{2,3}}(-\widetilde{\tau}^* Z_1^2) = 0\) for \(i>0\). We refer to [7] for more details on \(B^k\) and \(B^{k-1,k}\).

3 Cohomology groups of symmetric powers of tautological bundles↩︎

In this section, we prove Theorem 1. The main ingredient is the description of higher direct images \(R^i \alpha_{k,*} \mathcal{O}_{B^k}\). The following proposition is a higher dimensional generalization of [3], which deals with the case where \(X\) is a curve. When \(i=0\), we have \(\alpha_{k,*} \mathcal{O}_{B^k} = \mathcal{O}_{\sigma_k}\) for \(k=2\) and \(3\) by Theorem 4 \((1)\).

We are ready to give the proof of Theorem 1.

4 Hilbert polynomials of secant varieties↩︎

This section is devoted to the proof of Theorem 2. Throughout the section, we keep assuming that \(L\) is a sufficiently ample line bundle on a smooth projective complex variety \(X\) of dimension \(n\).

4.1 Hilbert polynomial of \(\sigma_2\)↩︎

We want to compute \(P_{\sigma_2}(\ell) = \chi(\mathcal{O}_{\sigma_2}(\ell))\) for \(\ell \in \mathbb{Z}\). We begin with the short exact sequence \[0 \longrightarrow \mathscr{I}_{X/\sigma_2} \longrightarrow \mathcal{O}_{\sigma_2} \longrightarrow \mathcal{O}_{X} \longrightarrow 0.\] We find \[\chi( \mathcal{O}_{\sigma_2}(\ell)) = \chi(\mathcal{O}_X(\ell)) + \chi (\mathscr{I}_{X/\sigma_2}(\ell)).\] By Theorem 4 (3), we have \[\chi (\mathscr{I}_{X/\sigma_2}(\ell)) = \chi(\mathcal{O}_{B^2}(\ell H_2 - Z_1^2)).\] Recall that \(\mathcal{O}_{B^2}(-Z_1^2) = \mathcal{O}_{B^2}(-2) \otimes \pi_2^* A_{2,L}\). For an integer \(i \geq 1\), we have a short exact sequence \[0 \longrightarrow \mathcal{O}_{B^2}(\ell H_2-(i+1)Z_1^2) \longrightarrow \mathcal{O}_{B^2}(\ell H_2 -iZ_1^2) \longrightarrow \mathcal{O}_{Z_1^2}(\ell H_2 -iZ_1^2). \longrightarrow 0.\] Now, assume \(\ell = 2m+1\) for some integer \(m \geq 0\). As \(\mathcal{O}_{B^2}( \ell H_2 - (m+1)Z_1^2) = \mathcal{O}_{B^3}(-1) \otimes \pi_2^* A_{2,L}^{m+1}\), we get \(\chi( \mathcal{O}_{B^2}( \ell H_2 - (m+1)Z_1^2) )=0\). Considering the above short exact sequences for \(i=1, \ldots, m\), we see that \[\chi(\mathcal{O}_{B^2}(\ell H_2 - Z_1^2)) = \sum_{i=1}^m \chi (\mathcal{O}_{Z_1^2}(\ell H_2 -iZ_1^2) ).\] Recall that \(Z_1^2 = X^{[1,2]}\). Then we may write \[\mathcal{O}_{Z_1^2}(\ell H_2 -iZ_1^2) = \rho^* A_{2,L}^i \otimes \mathrm{res}^* L^{\ell - 2i} = (\tau^* L^i \otimes \mathrm{res}^* L^{\ell-i})(-2iF_1).\] By Lemma 8 shown below, we have \[\chi\big( (\tau^* L^i \otimes \mathrm{res}^* L^{\ell-i})(-2iF_1) \big) = \chi( L^{\ell-i}) \cdot \chi(L^i) - \sum_{j=0}^{2i-1} \chi(S^j \Omega_X \otimes L^{\ell}).\] Combining all together, we obtain \[\chi(\mathcal{O}_{\sigma_2}(2m+1)) = \chi(L^{2m+1}) + \sum_{i=1}^{m} \left(\chi( L^{2m-i+1}) \cdot \chi(L^i) - \sum_{j=0}^{2i-1} \chi(S^j \Omega_X \otimes L^{2m+1}) \right)\] for \(m \geq 0\). Note that \(P_{\sigma_2}(\ell) = \chi(\mathcal{O}_{\sigma_2}(\ell))\) is a polynomial in \(\ell\) of degree \(2n+1 = \dim \sigma_2\). By Lagrange interpolation, \[P_{\sigma_2}(\ell) = \sum_{m=0}^{2n+1} P_{\sigma_2}(2m+1) \prod_{\substack{t=0 \\ t \neq m}}^{2n+1} \frac{\ell - 2t-1}{2m - 2t}\] for any integer \(\ell \in \mathbb{Z}\). To finish the proof of Theorem 2 \((1)\), it only remain to show the following lemma.

4.2 Hilbert polynomial of \(\sigma_3\)↩︎

We want to compute \(P_{\sigma_3}(\ell) = \chi(\mathcal{O}_{\sigma_3}(\ell))\) for \(\ell \in \mathbf{Z}\). The basic strategy is the same as in the case of \(\sigma_2\), but the computation becomes considerably more complicated. Rather than presenting every detailed step of the calculation, we focus on the essential ideas and key steps of the argument. The omitted part should be manageable for the reader. As before, we start by considering the short exact sequence \[0 \longrightarrow \mathcal{I}_{\sigma_2/\sigma_3} \longrightarrow \mathcal{O}_{\sigma_3} \longrightarrow \mathcal{O}_{\sigma_2} \longrightarrow 0.\] We find \[\chi(\mathcal{O}_{\sigma_3}(\ell)) = \chi (\mathcal{O}_{\sigma_2}(\ell)) + \chi(\mathscr{I}_{\sigma_2/\sigma_3}(\ell)) = P_{\sigma_2}(\ell) + \chi(\mathscr{I}_{\sigma_2/\sigma_3}(\ell)).\] By Theorem 4 (3), we have \[\chi(\mathscr{I}_{\sigma_2/\sigma_3}(\ell)) = \chi(\mathcal{O}_{B^3}(\ell H_3 - Z_2^3)).\] Recall that \(\mathcal{O}_{B^3}(-Z_2^3) = \mathcal{O}_{B^3}(-3) \otimes \pi_3^* A_{3,L}\). For an integer \(i \geq 1\), we have a short exact sequence \[0 \longrightarrow \mathcal{O}_{B^3}(\ell H_3-(i+1)Z_2^3) \longrightarrow \mathcal{O}_{B^3}(\ell H_3 -i Z_2^3 ) \longrightarrow \mathcal{O}_{Z_2^3}(\ell H_3 - iZ_2^3) \longrightarrow 0.\] Now, assume \(\ell = 3m+2\) for some integer \(m \geq 0\). As \(\mathcal{O}_{B^3}(\ell H_3 - (m+1)Z_2^3) = \mathcal{O}_{B^3}(-1) \otimes \pi_3^* A_{3,L}^{m+1}\), we get \(\chi(\mathcal{O}_{B^3}(\ell H_3 - (m+1)Z_2^3) ) = 0\). Considering the above short exact sequences for \(i=1, \ldots, m\), we see that \[\chi(\mathcal{O}_{B^3}(\ell H_3 - Z_2^3)) = \sum_{i=1}^m \chi(\mathcal{O}_{Z_2^3}(\ell H_3 - iZ_2^3)).\] Next, from the short exact sequence \[0 \longrightarrow \alpha_{2,3,*}\mathcal{O}_{B^{2,3}}(-\tilde{\tau}^*Z_1^2) \longrightarrow \mathcal{O}_{Z_2^3} \longrightarrow \mathcal{O}_{Z_1^3} \longrightarrow 0,\] we get \[\chi(\mathcal{O}_{Z_2^3}(\ell H_3 - iZ_2^3)) = \chi( \alpha_{2,3,*}\mathcal{O}_{B^{2,3}}(-\tilde{\tau}^*Z_1^2) \otimes \mathcal{O}_{Z_2^3}(\ell H_3 - iZ_2^3)) + \chi(\mathcal{O}_{Z_1^3}(\ell H_3 - iZ_2^3)).\] Note that \[\begin{align} & \chi( \alpha_{2,3,*}\mathcal{O}_{B^{2,3}}(-\tilde{\tau}^*Z_1^2) \otimes \mathcal{O}_{Z_2^3}(\ell H_3 - iZ_2^3))) = \chi \big( (\tau^* (S^{\ell - 3i-2} E_{2,L} \otimes A_{2,L}^{i+1}) \otimes \mathrm{res}^* L^i)(-2iF_2) \big);\\ & \chi(\mathcal{O}_{Z_1^3}(\ell H_3 - iZ_2^3)) = \chi\big( (\tau^* A_{2,L}^i \otimes \mathrm{res}^* L^{\ell - 2i})(-2iF_2) \big). \end{align}\] The Euler characteristics on the right-hand side are defined on \(X^{[2,3]}\). For an integer \(j \geq 0\), we have a short exact sequence \[0 \longrightarrow \mathcal{O}_{X^{[2,3]}}(-(j+1)F_2) \longrightarrow \mathcal{O}_{X^{[2,3]}}(-jF_2) \longrightarrow \mathcal{O}_{F_2}(-jF_2) \longrightarrow 0.\] Recall that there is the blow-up \(\textrm{bl}\colon X^{[2,3]} \to X \times X^{[2]}\) along \(\mathcal{Z}_2=X^{[1,2]}\) with exceptional divisor \(F_2\). Let \(N:=N_{\mathcal{Z}_2/X \times X^{[2]}}\) be the normal bundle of \(\mathcal{Z}_2\) in \(X \times X^{[2]}\). Considering the above short exact sequence for \(j=0,\ldots, 2i-1\) after tesnsoring by relevant vector bundles on \(X^{[2,3]}\), we see that \[\begin{align} & \chi \big( (\tau^* (S^{\ell - 3i-2} E_{2,L} \otimes A_{2,L}^{i+1}) \otimes \mathrm{res}^* L^i)(-2iF_2) \big) \\[-5pt] & = \chi(S^{\ell - 3i-2} E_{2,L} \otimes A_{2,L}^{i+1}) \cdot \chi(L^i) - \sum_{j=0}^{2i-1} \chi(S^j N^{\vee} \otimes \rho^* (S^{\ell-3i-2} E_{2,L} \otimes A_{2,L}^{i+1}) \otimes \mathrm{res}^* L^i);\\ & \chi\big( (\tau^* A_{2,L}^i \otimes \mathrm{res}^* L^{\ell - 2i})(-2iF_2) \big) = \chi(A_{2,L}^i) \cdot \chi(L^{\ell-2i}) - \sum_{j=0}^{2i-1}\chi(S^j N^{\vee} \otimes (\tau^* L^i \otimes \mathrm{res}^* L^{\ell-i})(-2iF_1)). \end{align}\] The Euler characteristics on the right-end terms are defined on \(X^{[1,2]}\). From the suitable symmetric powers of the short exact sequence \[0 \longrightarrow \tau^* L(-F_1) \longrightarrow \rho^* E_{2,L} \longrightarrow \mathrm{res}^*L \longrightarrow 0,\] we deduce that \[\chi(S^j N^{\vee} \otimes \rho^* (S^{\ell-3i-2} E_{2,L} \otimes A_{2,L}^{i+1}) \otimes \mathrm{res}^* L^i) = \sum_{k=0}^{\ell-3i-2} \chi\big(S^j N^{\vee} \otimes (\tau^* L^{k + i+1} \otimes \mathrm{res}^* L^{\ell-i-k-1})(-(k+2i+2)F_1)\big).\] Putting together all the results obtained so far, we have \[\begin{align} P_{\sigma_3}(3m+2) &= P_{\sigma_2}(3m+2) + \sum_{i=1}^m \Big(\chi(S^{3m - 3i} E_{2,L} \otimes A_{2,L}^{i+1}) \cdot \chi(L^i) + \chi(A_{2,L}^i) \cdot \chi(L^{3m-2i+2})\Big)\\[-5pt] & ~ - \sum_{i=1}^m \sum_{j=0}^{2i-1} \chi\big(S^j N^{\vee} \otimes (\tau^* L^i \otimes \mathrm{res}^* L^{3m-i+2})(-2iF_1)\big)\\[-5pt] & ~ - \sum_{i=1}^m \sum_{j=0}^{2i-1} \sum_{k=0}^{3m-3i} \chi\big(S^j N^{\vee} \otimes (\tau^* L^{k + i+1} \otimes \mathrm{res}^* L^{3m-i-k+1})(-(k+2i+2)F_1)\big). \end{align}\] for \(m \geq 0\). In the end, it remains to compute \(\chi(S^a E_{2,L} \otimes A_{2,L}^b)\) and \(\chi(S^a N^{\vee} \otimes (\tau^* L^b \otimes \mathrm{res}^* L^c)(-dF_1))\). Finally, Theorem 2 \((2)\) follows from the next two lemmas, which complete the remaining tasks, combined with Lagrange interpolation.

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