Proof. There is another standard way to compute the degree of the rational map $\varphi _V$ , in terms of Hilbert–Samuel multiplicities of $\mathcal {I}$ . The Hilbert–Samuel multiplicity of an ideal sheaf $\mathcal {I}$ at a point p is defined as $e_p(\mathcal {I}):=\mathrm {length}(\mathcal {O}_{S,p}/(f,g))$ , where $f,g\in \mathcal {I}\otimes \mathcal {O}_{S,p}$ are general elements. Then

$$\begin{align*}\mathrm{deg}(\varphi_V)=L^2-\sum_{p \in S}e_p(\mathcal{I}). \end{align*}$$

Now, by [Reference Huneke, Ma, Quy and SmirnovHMQS20, Proposition 5.7] $e_p(\mathcal {I})\le \frac {4}{3}\mathrm {colength}(\mathcal {I},p)$ (notice that $\mathcal {I}\otimes \mathcal {O}_{S,p}$ has at most three generators). The lemma follows.

Proof. The inequality follows from the previous lemma, together with the observation that $c_2(E)=L^2-\mathrm {colength}( \mathcal I)$ . For the second part, observe that under the assumption $\operatorname {\mathrm {Pic}}(S)=\mathbb {Z}\cdot L$ the kernel bundle $E^\vee $ is stable; otherwise, we would have $\mathcal {O}_S\hookrightarrow E^\vee $ which contradicts the inclusion $V\hookrightarrow H^0(L\otimes \mathcal {I})$ . Hence, by Mukai’s theory of stable vector bundles on $K3$ surfaces ([Reference MukaiMuk87, Corollary 2.5]) we have $v(E)^2\geq -2$ (see Subsection 2.3 for the definition of $v(E)$ ), which in this case reads as $c_2(E)\geq \lfloor \frac {g+3}{2} \rfloor $ .

Proof. We choose the local coordinates $x,y$ such that $\mathcal {I}_\xi =(x,y^m)$ locally at p. Given $e_1,e_2$ local generators of E at p, let $\mathcal {I}_j$ be the monomial ideal generated by monomials appearing in the j-th coordinate of sections of $V^\vee $ ; we may choose $e_1,e_2$ so that $x\in \mathcal {I}_1$ .

In this framework, three general elements $s_1,s_2,s_3$ of $V^\vee $ (giving a basis) have local expressions

$$\begin{align*}s_i=\left((x-f_i(y))\cdot t_i(x,y),u_i(x,y)\right), \end{align*}$$

where $f_i(y)\in (y^m)$ and $t_i(x,y)\in \mathcal {O}_{S,p}^*$ (by the implicit function theorem).

Of course, one has $u_i(f_i(y),y)\in (y^f)$ . Actually, since $Z_{cycle}(s_1+\lambda s_2+\mu s_3)$ has degree $\geq f$ at p, one has

$$\begin{align*}u_1(f_{\lambda,\mu}(y),y)+\lambda u_2(f_{\lambda,\mu}(y),y)+\mu u_3(f_{\lambda,\mu}(y),y)\in (y^f) \end{align*}$$

for every $\lambda ,\mu $ and every expression $(x-f_1)t_1+\lambda (x-f_2)t_2+\mu (x-f_3)t_3=(x-f_{\lambda ,\mu }(y))\cdot t_{\lambda ,\mu }(x,y)$ with $f_{\lambda ,\mu }\in (y^m)$ and $t_{\lambda ,\mu }\in \mathcal {O}_{S,p}^*$ .

This last condition implies that, after replacing $e_2$ by an appropriate combination of $e_1$ and $e_2$ , one may assume that $x^ky^l\notin \mathcal {I}_2$ as long as $f>mk+l$ . Therefore, for the new choice of local generators one has $\mathcal {I}_2\subset (x^{\lceil \frac {f}{m}\rceil },x^{\lceil \frac {f-1}{m}\rceil }y,...,x^{\lceil \frac {1}{m}\rceil }y^{f-1},y^f)$ , which proves the assertion.

Proof. By applying the functor $\operatorname {\mathrm {Hom}}_{\mathcal {O}_X}(-,k(p))$ to the short exact sequence

$$\begin{align*}0\longrightarrow \mathcal{I}_{T/X}\longrightarrow \mathcal{O}_X\longrightarrow \mathcal{O}_T\longrightarrow 0, \end{align*}$$

we get an isomorphism $\operatorname {\mathrm {Hom}}_{\mathcal {O}_X}(\mathcal {I}_{T/X},k(p))\cong \operatorname {\mathrm {Ext}}^1_{\mathcal {O}_X}(\mathcal {O}_T,k(p))$ . Now, the claim follows from the adjunction $\operatorname {\mathrm {Hom}}_{\mathcal {O}_X}(\mathcal {I}_{T/X},k(p))\cong \operatorname {\mathrm {Hom}}_{k}(\mathcal {I}_{T/X}\otimes k(p),k(p))$ and Nakayama’s lemma.

Proof. Since $h^2(E\otimes \mathcal {I}_\xi )=0$ for every $E\in \mathcal {M}$ and $\xi \in S^{[n]}$ , we have an isomorphism

$$\begin{align*}R^1\Phi_{\mathcal{E}}(\mathcal{I}_\xi)\otimes k(E)\overset{\cong}{\longrightarrow} H^1(E\otimes\mathcal{I}_\xi) \end{align*}$$

by cohomology and base change (which is still valid in the twisted setting, since it is a local property). Only the last assertion is not an immediate consequence of this isomorphism.

To complete the proof, note that all elements of $\overline {JC_\xi }$ are ideal sheaves, up to twist by a line bundle (see, e.g., [Reference D’SouzaD’S79, Proposition 3.2]). Since the curve $C_\xi $ is Gorenstein (it lies on a smooth surface), the minimal number of generators of an ideal sheaf at a point is bounded by the multiplicity of the curve at that point, which finishes the proof.

Proof. Let $E\in \mathcal {M}$ . Note that $G\in \operatorname {\mathrm {Coh}}^\beta (S)$ for all $\beta <\frac {-2}{5}$ , and it is $\sigma _{\alpha ,\beta }$ -stable for all $\alpha $ if $\beta =\frac {-1}{2},\frac {-3}{7}$ ; in particular, G and $E^\vee [1]$ are $\sigma _0$ -stable of the same slope. It follows that $\hom (E^\vee [1],G)=0=\hom (G,E^\vee [1])$ , which gives $\operatorname {\mathrm {ext}}^1(E^\vee [1],G)=\langle v(E^\vee [1],v(G)\rangle =1$ .

Therefore, there is a (unique) nontrivial extension

(3.1) $$ \begin{align} 0\to G\to \varphi(E)\to E^\vee[1]\to 0, \end{align} $$

and $\varphi (E)$ must be $\sigma $ -semistable for $\sigma \in \{\beta =\frac {-3}{7}\}$ above $\sigma _0$ . Otherwise, for a $\sigma $ -destabilizing subobject $R\subset \varphi (E)$ the composition $R\hookrightarrow \varphi (E)\twoheadrightarrow E^\vee [1]$ would be nonzero (since G is $\sigma $ -stable), and $\sigma _0$ -stability of $E^\vee [1]$ would force $R\cong E^\vee [1]$ , which splits the extension (3.1).

By $\sigma $ -stability of $\varphi (E)$ , it follows that $\varphi (E)\in \mathcal {M}(3,-L,2)$ , and since this construction can be carried out in familiesFootnote ², the assignment $\varphi $ is a morphism of schemes. Furthermore, the sequence (3.1) destabilizes $\varphi (E)$ along $W\cap \{\beta <\frac {-2}{5}\}$ : G and $E^\vee [1]$ are the factor of the HN filtration of $\varphi (E)$ for stability conditions along $\beta =\frac {-3}{7}$ that are below $\sigma _0$ .

By rotating the distinguished triangle, we obtain a short exact sequence

$$\begin{align*}0\to \varphi(E)\to E^\vee[1]\to G[1]\to 0 \end{align*}$$

in $\operatorname {\mathrm {Coh}}^\beta (S)$ ( $\frac {-2}{5}\leq \beta <\frac {-1}{3}$ ), which destabilizes $E^\vee [1]$ along $W\cap \{\beta>\frac {-2}{5}\}$ . The HN factors of $E^\vee [1]$ for stability conditions just below $W\cap \{\beta =\frac {-3}{8}\}$ will be precisely $\varphi (E)$ and $G[1]$ .

On the other hand, starting with $F\in \mathcal {M}(3,-L,2)$ , one can construct an element $\psi (F)\in \mathcal {M}$ arising as the (unique) nontrivial extension

$$\begin{align*}0\to F\to \psi(F)^\vee[1]\to G[1]\to 0 \end{align*}$$

(for the uniqueness of such an extension, we use again the stability of F and $G[1]$ along $\beta =\frac {-3}{8}$ ). By rotating the triangle, we can destabilize F along $W\cap \{\beta <\frac {-2}{5}\}$ .

The morphisms $\varphi $ and $\psi $ are inverse to each other, as easily follows from the uniqueness of the HN filtrations of $E^\vee [1]$ , $\varphi (E)$ , F and $\psi (F)^\vee [1]$ just below W. This concludes the proof.

Proof. Assume that $h^1(E\otimes \mathcal {I}_\xi )\geq 1$ , that is, there exists a short exact sequence

$$\begin{align*}0\to F\to \mathcal{I}_\xi\to E^\vee[1]\to 0 \end{align*}$$

destabilizing $\mathcal {I}_\xi $ along W (with $F\in \mathcal {M}(3,-L,2)$ ). Consider the short exact sequence

(3.2) $$ \begin{align} 0\to G\to F\to \psi(F)^\vee[1]\to 0 \end{align} $$

destabilizing F along $W\cap \{\beta <\frac {-2}{5}\}$ , constructed in the proof of Lemma 3.1. Then the HN factors of $\mathcal {I}_\xi $ (for stability conditions just below $\sigma _0$ ) are G and T, where T is the extension of $E^\vee [1]$ by $\psi (F)^\vee [1]$ obtained as the image of $\mathcal {I}_\xi $ under the induced map

$$\begin{align*}\operatorname{\mathrm{Ext}}^1(E^\vee[1],F)\longrightarrow\operatorname{\mathrm{Ext}}^1(E^\vee[1],\psi(F)^\vee[1]). \end{align*}$$

If $E\neq \psi (F)$ , of course we have $T=E^\vee [1]\oplus \psi (F)^\vee [1]$ ; otherwise, T is a nontrivial extension. Indeed, if $E= \psi (F)$ , then the above map of Ext-groups is an isomorphism (as follows from the long exact sequence obtained by applying the functor $\operatorname {\mathrm {Hom}}(E^\vee [1],-)$ to the sequence (3.2)).

In any case, since $\operatorname {\mathrm {Hom}}(G,E^\vee [1])=0$ (recall that G are $E^\vee [1]$ are $\sigma _0$ -stable of the same slope), we obtain that $\hom (\mathcal {I}_\xi , E^\vee [1])=\hom (T,E^\vee [1])=1$ .

Proof. We argue by contradiction. Suppose that there exists $V^\vee \in \mathrm {Gr}(3,H^0(E))$ such that $\bigcap _{s\in V^\vee }Z_{cycle}(s)=2p$ and $\bigcap _{s \in V^\vee }Z(s)=\{p\}$ . According to Lemma 2.4, we can choose local coordinates and local generators $e_1,e_2$ of E at p, such that the subsheaf $E_1 \subset E$ defined by ${E_1}|_{S\setminus p}=E|_{S\setminus p}$ and $E_1=m_p e_1+m_p^2e_2$ in a neighborhood of p has $V^\vee \subset H^0(E_1)$ , in particular we get $h^0(E_1)\ge 3$ . Since the quotient $E/E_1$ is a $4$ -dimensional vector space, we get $\chi (E_1)=1$ and hence $h^1(E_1)=2$ . Let us consider the double extension

It is easy to compute that $h^0(E_1 \otimes k(p))=5$ ( $E_1$ has $xe_1$ , $ye_1$ , $x^2e_2$ , $xye_2$ , $y^2e_2$ as local generators), hence F cannot be locally free at p. We get $c_2(F^{**})<c_2(F)=c_2(E_1)=9$ . It suffices to prove that $F^{**}$ must be stable since there exists no stable rank $4$ vector bundle with these invariants. To this end, consider the commutative diagram with exact rows

where $V^\vee \in \mathrm {Gr}(3,H^0(E_1))$ . The map $W\to H^0(L)$ is injective; indeed, $V \to H^0(L)$ is injective (by stability of E) and the extension defining F is nonsplit (none of the maps $\mathcal O_S\to F$ in the definition of F split). It follows that $H^0(F^*)=0$ , which is enough to conclude the stability of the kernel bundle $F^*$ since $\operatorname {\mathrm {Pic}}(S)=\mathbb {Z}\cdot L$ (if T is a vector bundle destabilizing $F^*$ , then T must be of the form $\mathcal {O}_S^{\oplus \operatorname {\mathrm {rk}} T}$ since $T\hookrightarrow W\otimes \mathcal {O}_S$ , contradiction).

Proof. By Corollary 2.3 the only other possibility is $c_2(E)=6$ and $V^\vee \in \mathrm {Gr}(3,H^0(E))$ with $\bigcap _{s\in V^\vee }Z_{cycle}(s)$ of degree $2$ .

If $\bigcap _{s\in V^\vee }Z(s)$ is a reduced point, then by Lemma 2.4 in an appropriate trivialization of E near p all sections $s\in V^\vee $ are of the form $s=ae_1+be_2$ , with $a\in m_p\setminus m_p^2$ and $b\in m_p^2$ . Since $V=\bigwedge ^2 V^\vee $ , it follows that the base ideal $\mathcal {I}$ satisfies $\mathcal {I}\subset \mathcal {I}_p^3$ ; therefore, $\mathrm {colength}(\mathcal {I})\geq 7$ (because $\mathcal {I}$ is generated by three elements). This implies that $c_2(E)=L^2-\mathrm {colength}(\mathcal {I})\leq 5$ .

Hence, if $c_2(E)=6$ , we must have $h^0(E \otimes \mathcal I_\xi )\ge 3$ for some $\xi \in S^{[2]}$ . Consider now the subsheaf $E_1 \subset E$ generated by $E \otimes \mathcal I_\xi $ and $\mathrm {Im}(\mathcal O_S \otimes H^0(E) \to E)$ . The quotient $E_1/E\otimes \mathcal I_\xi $ is of length at most $2$ , since it is a principal $\mathcal {O}_\xi $ -submodule in $E/E \otimes \mathcal I_\xi $ (it is generated by a section in $H^0(E)\setminus H^0(E \otimes \mathcal I_\xi )$ ). This implies $\chi (E_1)\leq \chi (E \otimes \mathcal I_\xi )+2=2$ , and since $h^0(E_1)=4$ , we obtain $h^1(E_1)\ge 2$ . Now, the nontrivial extension

gives a stable torsion free sheaf F with Mukai vector $v(F)=(h^1(E_1)+2,L,2)$ , which yields a contradiction if $h^1(E_1)\geq 2$ .

Proof. Let us define a morphism $\gamma :\mathcal {M}^{[2]}\longrightarrow S^{[2]}$ as follows.

Given a reduced element of $\mathcal {M}^{[2]}$ , supported at two distinct $E_1,E_2\in \mathcal {M}$ , we consider the object M fitting into an extension

$$\begin{align*}0\to G\to M\to E_1^\vee[1]\oplus E_2^\vee[1]\to 0 \end{align*}$$

such that the induced elements of $\operatorname {\mathrm {Ext}}^1(E_i^\vee [1],G)$ are nonzero. Due to the action of automorphisms of $E_1^\vee [1]\oplus E_2^\vee [1]$ , such an object M is unique up to isomorphism.

On the other hand, given a nonreduced element of $\mathcal {M}^{[2]}$ supported at $E\in \mathcal {M}$ with tangent direction $T\in \mathbb {P}(\operatorname {\mathrm {Ext}}^1(E,E))$ , we consider the corresponding $\widetilde {T}:=T^\vee [1]\in \mathbb {P}(\operatorname {\mathrm {Ext}}^1(E^\vee [1],E^\vee [1]))$ and take the object M fitting into

$$\begin{align*}0\to G\to M\to \widetilde{T}\to 0 \end{align*}$$

such that the induced extension in $\operatorname {\mathrm {Ext}}^1(E^\vee [1],G)$ is nonzero. Again, such an object M is unique up to isomorphism (due to the action of automorphisms of $\widetilde {T}$ ).

Let us assume for a moment that, in both cases, the object M is of the form $M=\mathcal {I}_\xi $ for some $\xi \in S^{[2]}$ (we will prove it below). Then this rule defines the morphism $\gamma $ , which is injective on closed points (indeed, given $M=\mathcal {I}_\xi $ in the image of $\gamma $ , the corresponding element of $\mathcal {M}^{[2]}$ can be recovered from the last HN factor of M just below $\sigma _0$ ). Therefore, $S^{[2]}$ equals the image of $\gamma $ (both $\mathcal {M}^{[2]}$ and $S^{[2]}$ are projective of the same dimension). Since $\gamma $ is bijective on closed points and $S^{[2]}$ is normal, it follows that $\gamma $ is an isomorphism.

Hence, it only remains to prove that the objects M constructed above are of the form $M=\mathcal {I}_\xi $ , $\xi \in S^{[2]}$ . Since there are no walls above W for the Mukai vector $(1,0,-1)$ , it suffices to check that M is $\sigma '$ -semistable for some stability condition $\sigma '$ above W. We will assume that M comes from a nonreduced element of $\mathcal {M}^{[2]}$ (the reduced case being similar but simpler).

Let $\sigma '$ be close enough to $\sigma _0$ (above W), and assume that M is not $\sigma '$ -semistable. Since M is $\sigma _0$ -semistable, any destabilizing subobject $A\subset M$ is $\sigma _0$ -semistable of the same slope as M.

The composition $G\hookrightarrow M\twoheadrightarrow M/A$ is either zero or injective since G is $\sigma _0$ -stable of the same slope as $M/A$ . In the first case, we have $G\hookrightarrow A$ and hence $A/G\hookrightarrow \widetilde {T}$ . Consider the composition $A/G\hookrightarrow \widetilde {T}\twoheadrightarrow E^\vee [1]$ . By $\sigma _0$ -stability of $E^\vee [1]$ , this morphism is either zero or surjective. If $A/G\to E^\vee [1]$ is zero then either $A/G=0$ or $A/G\cong E^\vee [1]=\ker (\widetilde {T}\twoheadrightarrow E^\vee [1])$ (hence $\nu _{\sigma '}(A)<\nu _{\sigma '}(M)$ , contradiction); whereas if $A/G\hookrightarrow \widetilde {T}\twoheadrightarrow E^\vee [1]$ is surjective, then either $A/G=\widetilde {T}$ (i.e., $A=M$ , contradiction) or the extension $0\to E^\vee [1]\to \widetilde {T}\to E^\vee [1]\to 0$ splits.

On the other hand, if $G\hookrightarrow M/A$ is injective, one easily checks that $A\hookrightarrow \widetilde {T}$ . Now, we consider the composition $A\hookrightarrow \widetilde {T}\twoheadrightarrow E^\vee [1]$ , which again will be either zero or surjective. If it is surjective, then either $A\cong \widetilde {T}$ (which splits $0\to G\to M\to \widetilde {T}\to 0$ ) or $A\cong E^\vee [1]$ (which splits $0\to E^\vee [1]\to \widetilde {T}\to E^\vee [1]\to 0$ ). If $A\hookrightarrow \widetilde {T}\twoheadrightarrow E^\vee [1]$ is zero, then $A\cong E^\vee [1]=\ker (\widetilde {T}\twoheadrightarrow E^\vee [1])$ and the induced element in $\operatorname {\mathrm {Ext}}^1(E^\vee [1],G)$ is zero, which gives again a contradiction.

Proof. This follows from the uniqueness of the HN filtration of $\mathcal {I}_\xi $ just below $\sigma _0$ .

Proof. The isomorphism is just a reformulation of Lemma 2.13. For the second part, observe that the nonemptiness of $R_3$ implies that some plane quartic of the linear system $|\hat {L}|$ has a triple point, hence it is rational. If $(S,L)$ is very general, then $(\mathcal {M},\hat L)$ must be very general among quartic surfaces; but in that case a theorem of Chen [Reference ChenChe02] ensures that all the rational curves in the linear system $|\hat {L}|$ must be nodal, which gives a contradiction.

Proof. If $\xi $ is a reduced point p, then by Lemma 2.4 we have $s_p\in m_p\cdot e_1+m_p^3\cdot e_2$ for the local equations of $s\in V^\vee $ in an appropriate trivialization $E_p\cong e_1\cdot \mathcal {O}_{S,p}\oplus e_2\cdot \mathcal {O}_{S,p}$ . In view of the isomorphism $V\cong \bigwedge ^2V^\vee $ , it follows that the base ideal $\mathcal {I}$ satisfies $\mathcal {I}\otimes \mathcal {O}_{S,p}\subset m_p^4$ locally at p; this implies that the degree of the map is $\leq L^2-16=0$ , contradiction.

Now, assume that $\xi $ is not reduced, locally of the form $(x,y^2)$ at $p=\mathrm {Supp}(\xi )$ . In that case, by Lemma 2.4 we have $s_p\in (x,y^2)\cdot e_1+(x^2,xy,y^{3})\cdot e_2$ for all $s\in V^\vee $ in an appropriate trivialization $E_p\cong e_1\cdot \mathcal {O}_{S,p}\oplus e_2\cdot \mathcal {O}_{S,p}$ of E near p. By the isomorphism $V\cong \bigwedge ^2V^\vee $ , it follows that the base ideal $\mathcal {I}$ satisfies $\mathcal {I}\otimes \hat {\mathcal {O}}_{S,p}\subset (x^3,x^2y, xy^{3},y^{5})$ locally at p. Furthermore, since $\mathcal {I}$ has at most three local generators at p, we obtain $\mathrm {colength}(\mathcal {I})\ge 11$ , contradiction.

Proof. In view of Lemma 4.3 and Lemma 4.5, the only possibility for a degree 3 map is that $\bigcap _{s\in V^\vee } Z(s)$ is reduced, consisting of two distinct points $p,q$ , and $\bigcap _{s\in V^\vee } Z_{cycle}(s)=2p+q$ . A new application of Lemma 2.4 yields $V\cong \bigwedge ^2 V^\vee \subset H^0(L\otimes \mathcal {I}_p^3\otimes \mathcal {I}_q^2)$ . We deduce $h^1(L\otimes \mathcal {I}_p^3\otimes \mathcal {I}_q^2)=2$ , hence we may consider the stable extension

It is easy to see that G is not locally free at p (G has rank 3, whereas $L \otimes \mathcal {I}_p^3\otimes \mathcal {I}_q^2$ has four local generators). Hence, $G\subset G^{**}$ is a nontrivial subsheaf, in particular $c_2(G^{**})<c_2(G)=9$ , whereas on the other hand we have $c_2(G^{**})\ge 8$ since $G^{**}$ is stable (stability follows from the fact that $G^*$ is a kernel bundle, with a proof similar to the stability of $F^*$ in the proof of Lemma 3.3). This forces $G^{**}$ to be the spherical vector bundle with $v(G^{**})=(3,L,3)$ .

Moreover, $G^{**}$ fits in a commutative diagram with exact rows and columns

where $\mathcal J$ is an ideal of colength $8$ such that $\mathcal J/\mathcal I_p^3\otimes \mathcal I_q^2\cong k(p)$ , hence it must have $3$ local generators at both p and q. This implies that the map $\mathcal O_S^{\oplus 2}\to G^{**}$ vanishes at p and q, hence $h^0(G^{**}\otimes \mathcal I_{p,q})\ge 2$ which is a contradiction to Lemma 4.2. The proof is complete.

Proof. First, let us show that $R_2 \not \subset S_{nred}^{[2]}$ . Assume the contrary. If $\pi _S(\mathbb {P}^1_E)$ is a point for every $E\in \mathcal {M}$ , then we get an injective map $\mathcal {M}\to S$ which is a contradiction. Otherwise, $\pi _S(\mathbb {P}^1_E)$ defines a rational curve on S moving with E, contradicting the rigidity of rational curves on S.

Now, let us show that the intersection is nonempty (then it will be automatically two-dimensional). For the general $E\in \mathcal {M}$ , consider the degree 2 map $D_E\to \mathbb {P}^1_E$ , where

$$\begin{align*}D_E:=\{(\xi,p)\in \mathbb{P}^1_E\times S\;|\;p\in\operatorname{\mathrm{supp}}(\xi)\}. \end{align*}$$

We claim that this degree 2 map must have a branch point. If $D_E$ is irreducible, this follows from Hurwitz formula. Otherwise, $D_E$ consists of two rational components; thus the image of the natural map $D_E\to S$ produces two rational curves on S. Since rational curves on S are rigid, it follows that this image (moving E) is a fixed curve C with (at most) two rational components. But the locus of length 2 subschemes supported at C is two-dimensional, hence $R_2$ can’t be contained in this locus (recall that $R_2$ is three-dimensional), which gives the contradiction.

Now, consider a component $W_S\subset R_2\cap S^{[2]}_{nred}$ such that the projection $\pi _{\mathcal {M}}:W_S\to \mathcal {M}$ is surjective. This exists, since we just proved that $D_E$ is irreducible for $E\in \mathcal {M}$ general (hence $\mathbb {P}^1_E$ contains a nonreduced subscheme). The general fiber of the map $\pi _S:W_S\to S$ consists of nonreduced subschemes supported at a general point p, hence it is contained in a rational curve. We deduce that if the general fiber of $\pi _S$ is positive-dimensional, then $W_S$ is covered by rational curves (the fibers of $\pi _S$ ); hence, $\mathcal {M}$ is covered by rational curves (recall that $W_S\to \mathcal {M}$ is dominant), contradiction. It follows that $W_S\to S$ is surjective, as desired.

Proof. For any (nonreduced) $\xi \in W_S$ , we have an associated E with $h^1(E\otimes \mathcal {I}_\xi )=1$ . We may consider the extension

$$\begin{align*}0 \longrightarrow E^\vee \longrightarrow F \longrightarrow \mathcal{I}_\xi \longrightarrow 0 \end{align*}$$

and the subsheaf $E_1:=\mathrm {Im}(F^\vee \to E)\subset E$ . By numerical reasons, we get $\mathrm {length}(E/E_1)=c_2(F^\vee )-c_2(E)=2$ , and also $E\otimes \mathcal {I}_\xi \subset E_1$ ; indeed, by applying the functor ${\mathcal {H}\mathit {om}}(-,\mathcal {O}_S)$ to the sequence above we get that $E_1$ is the kernel of $E\to {\mathcal {E}\mathit {xt}}^1(\mathcal {I}_\xi ,\mathcal {O}_S)=\mathcal {O}_\xi $ . Moreover, if $p=\operatorname {\mathrm {supp}}(\xi )$ , then $E_1\otimes k(p)$ is three-dimensional since $E_1$ is a quotient of a rank 3 locally free sheaf. It is then easy to deduce that $(E_1)_p=e_1\cdot \mathcal {O}_{S,p}+e_2 \cdot \mathcal {I}_\xi $ for some $e_1,e_2$ local generators of $E_p$ .

Note that $h^1(E_1)=1$ (since $h^1(F^\vee )=h^2(F^\vee )=0$ ). It follows that the subsheaf $\tilde E\subset E_1$ defined as $\tilde E_p=e_1\cdot m_p+e_2\cdot m_p^2$ around p (and $\tilde E_q=E_q$ for $q\neq p$ ) satisfies $h^1(\tilde E)\ge 1$ (since $\tilde E\subset E_1$ ). Actually, a quick numerical check gives $h^1(\tilde E)= 1$ and $\chi (\tilde E)=2$ , hence $h^0(\tilde E)=3$ . The subspace $H^0(\tilde E)\subset H^0(E)$ induces a map of degree $\leq 4$ by Proposition 2.1 (all sections are vanishing with order $2$ at p).

The corresponding morphism $W_S\to W^2_4(S,L)$ has image of dimension $2$ , since $\pi _{\mathcal {M}}:W_S \to \mathcal {M}$ is surjective and equals the composition

$$\begin{align*}W_S \longrightarrow W^2_4(S,L)_6 \longrightarrow \mathcal{M}, \end{align*}$$

where $W^2_4(S,L)_6$ parametrizes maps of degree $\le 4$ having kernel bundle with $c_2=6$ (this is the notation in the preliminaries), and the right arrow is the forgetful map $(E,V^\vee )\mapsto E$ .

Proof. Recall that a rational map $S\dashrightarrow \mathbb P^2$ induced by the primitive linear system is completely determined by the couple $(E,V^\vee )$ . We have $(E,V^\vee )\in R_2$ if and only if $V^\vee =H^0(E\otimes \mathcal {I}_\zeta )$ for some $\zeta \in S^{[2]}$ . Assume that $H^0(\tilde E)=H^0(E\otimes \mathcal {I}_\zeta )$ , where (following the notation of Lemma 4.8) $\tilde E$ is constructed from a nonreduced $\xi \in W_S$ .

If $\zeta $ is reduced, then all sections of $H^0(E\otimes \mathcal {I}_\zeta )$ vanish with order $2$ at the point where $\xi $ is supported. Hence, $\bigcap _{s\in V^\vee }Z_{cycle}(s)$ is of degree $\geq 3$ , which by Proposition 2.1 implies that $(E,V^\vee )\in W^2_3(S,L)$ , in contradiction to Corollary 4.6.

Now, suppose that $\zeta $ is nonreduced. We get that $\zeta $ is supported at the point p where $\tilde E$ is not locally free. This implies that all the sections in $V^\vee $ are locally contained in the subsheaf $e_1\cdot \mathcal {I}_\zeta +e_2\cdot m_p^2$ of E; in particular, we get $h^0(E\otimes \mathcal {I}_p^2)\ge 2$ . In order to finish the proof, it suffices to prove the following claim: For a general point $p\in S$ , there is no vector bundle $E_p\in \mathcal {M}$ such that $h^0(E_p\otimes \mathcal {I}_p^2)=h^1(E_p\otimes \mathcal {I}_p^2)=2$ .

Observe that such a bundle $E_p\in \mathcal {M}$ is a singular point of the curve

$$\begin{align*}C_{\mathcal{I}_p^2}=\{E\in \mathcal{M} \;|\; h^1(E\otimes \mathcal{I}_p^2)\ge 1 \}, \end{align*}$$

which has arithmetic genus $3$ (recall Lemma 2.13). Therefore, if the claim is false one can consider the rational map $S\dashrightarrow \mathcal {M}$ sending a general $p\in S$ to the unique singular point of $C_{\mathcal {I}_p^2}$ (note that if $C_{\mathcal {I}_p^2}$ has at least two singular points for every $p\in S$ , then $\{C_{\mathcal {I}_p^2}\}_{p\in S}$ defines a two-dimensional family of curves on $\mathcal {M}$ with geometric genus $\leq 1$ , contradiction). This rational map actually extends to a morphism $S\to \mathcal {M}$ ; indeed, the finite birational morphism

$$\begin{align*}\{(p,E)\in S\times\mathcal{M}\;|\;h^0(E\otimes\mathcal{I}_p^2)\geq 2\}\longrightarrow S \end{align*}$$

must be an isomorphism, by normality of S. Now, the morphism $S\to \mathcal {M}$ is étale by Hurwitz formula, which contradicts the fact that $\mathcal {M}$ is simply connected. This finishes the proof.

Proof. The proof is similar to that of Proposition 3.5. We define the morphism $\gamma $ as follows.

Given a reduced element of $\mathcal {M}^{[3]}$ , supported at $E_1,E_2,E_3\in \mathcal {M}$ , we consider the object M fitting into an extension

$$\begin{align*}0\to G\to M\to E_1^\vee[1]\oplus E_2^\vee[1]\oplus E_3^\vee[1]\to 0 \end{align*}$$

such that the induced elements in $\operatorname {\mathrm {Ext}}^1(E_i^\vee [1],G)$ are nonzero. Such an M is unique up to isomorphism (due to the action of automorphisms of $E_1^\vee [1]\oplus E_2^\vee [1]\oplus E_3^\vee [1]$ ), and of the form $M=\mathcal {I}_\xi $ for some $\xi \in S^{[3]}$ (indeed M is semistable above $\sigma _0$ , by arguing as in Proposition 3.5).

If our element of $\mathcal {M}^{[3]}$ consists of a tangent direction $T\in \mathbb {P}(\operatorname {\mathrm {Ext}}^1(E_1,E_1))$ at $E_1$ and a reduced point $E_2$ , we consider $\widetilde {T}:=T^\vee [1]\in \mathbb {P}(\operatorname {\mathrm {Ext}}^1(E_1^\vee [1],E_1^\vee [1]))$ and take the object M fitting into

$$\begin{align*}0\to G\to M\to \widetilde{T}\oplus E_2^\vee[1]\to 0 \end{align*}$$

such that the induced elements in $\operatorname {\mathrm {Ext}}^1(E_1^\vee [1],G)$ and $\operatorname {\mathrm {Ext}}^1(E_2^\vee [1],G)$ (the former via the map $\operatorname {\mathrm {Ext}}^1(\widetilde {T}\oplus E_2^\vee [1],G)\to \operatorname {\mathrm {Ext}}^1(\widetilde {T},G)\to \operatorname {\mathrm {Ext}}^1(E_1^\vee [1],G)$ ) are nonzero. Again, M is unique up to isomorphism and of the form $M=\mathcal {I}_\xi $ for some $\xi \in S^{[3]}$ .

Finally, given a tangent direction $T\in \mathbb {P}(\operatorname {\mathrm {Ext}}^1(E^\vee [1],E^\vee [1]))$ at $E\in \mathcal {M}$ , it is well known that all elements of $\mathcal {M}^{[3]}$ supported at E and containing T are parametrized by $\mathbb {P}(\operatorname {\mathrm {Ext}}^1(T,E))$ . Given such a $Q\in \mathbb {P}(\operatorname {\mathrm {Ext}}^1(T,E))$ , consider the corresponding $\widetilde {Q}:=Q^\vee [1]\in \mathbb {P}(\operatorname {\mathrm {Ext}}^1(E^\vee [1],T^\vee [1]))$ and take the object M sitting in an extension

$$\begin{align*}0\to G\to M\to \widetilde{Q}\to 0 \end{align*}$$

such that the induced element in $\operatorname {\mathrm {Ext}}^1(E^\vee [1],G)$ (via the map $\operatorname {\mathrm {Ext}}^1(\widetilde {Q},G)\to \operatorname {\mathrm {Ext}}^1(T^\vee [1],G)\to \operatorname {\mathrm {Ext}}^1(E^\vee [1],G)$ ) is nonzero. Again, M is unique and of the form $M=\mathcal {I}_\xi $ , for some $\xi \in S^{[3]}$ .

The morphism $\gamma $ is injective on closed points, and hence, arguing as in the proof of Proposition 3.5, we obtain that $\gamma $ is an isomorphism.

Proof of Proposition 5.1.

Given $\xi \in S^{[3]}$ and $E'\in \mathcal {M}$ , we want to determine the number $\hom (\mathcal {I}_\xi , E^{\prime \vee }[1])=h^1(E'\otimes \mathcal {I}_\xi )$ . To this end, we consider the HN filtration

$$\begin{align*}0\to G\to \mathcal{I}_\xi\to F\to 0 \end{align*}$$

of $\mathcal {I}_\xi $ for stability conditions just below $\sigma _0$ , established in Lemma 5.5. The object F is nothing but $F=\Phi _{\mathcal {E}}(\mathcal {O}_{\gamma ^{-1}(\xi )})^\vee [1]$ , where $\Phi _{\mathcal {E}}:\mathop {\mathrm {D}^{\mathrm {b}}}\nolimits (\mathcal {M})\to \mathop {\mathrm {D}^{\mathrm {b}}}\nolimits (S)$ is the Fourier–Mukai equivalence with kernel the universal bundle $\mathcal {E}$ on $S\times \mathcal {M}$ .

Due to the vanishing $\operatorname {\mathrm {Hom}}(G,E^{\prime \vee }[1])=0$ , we have equalities

$$\begin{align*}\begin{aligned} &\operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)}(\mathcal{I}_\xi, E^{\prime\vee}[1])=\operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)}(F, E^{\prime\vee}[1])=\\ &=\operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)}(\Phi_{\mathcal{E}}(\mathcal{O}_{\gamma^{-1}(\xi)})^\vee[1], \Phi_{\mathcal{E}}(k(E'))^\vee[1])=\operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathcal{M})}(k(E'),\mathcal{O}_{\gamma^{-1}(\xi)}). \end{aligned} \end{align*}$$

If $E'\notin \operatorname {\mathrm {supp}} \gamma ^{-1}(\xi )$ , it immediately follows that $\hom (\mathcal {I}_\xi , E^{\prime \vee }[1])=0$ .

On the other hand, if $E'\in \operatorname {\mathrm {supp}} \gamma ^{-1}(\xi )$ , then the equality

$$\begin{align*}\hom(k(E'),\mathcal{O}_{\gamma^{-1}(\xi)})=\operatorname{\mathrm{ext}}^1(k(E'),\mathcal{O}_{\gamma^{-1}(\xi)})-1=\operatorname{\mathrm{ext}}^1(\mathcal{O}_{\gamma^{-1}(\xi)},k(E'))-1 \end{align*}$$

holds; indeed, it is a consequence of the equalities $\operatorname {\mathrm {ext}}^2(k(E'),\mathcal {O}_{\gamma ^{-1}(\xi )})= \hom (\mathcal {O}_{\gamma ^{-1}(\xi )},k(E'))=1$ and $\chi (k(E'),\mathcal {O}_{\gamma ^{-1}(\xi )})=0$ .

Therefore, by applying Lemma 2.6 we obtain

$$\begin{align*}\hom(\mathcal{I}_\xi, E^{\prime\vee}[1])=\left\{ \begin{array}{@{}cl} 1 & \text{if }\gamma^{-1}(\xi)\text{ is curvilinear at }E'\\ 2 & \text{if }\gamma^{-1}(\xi)\text{ is not curvilinear at }E'.\\ \end{array} \right. \end{align*}$$

In particular, for a given $E\in \mathcal {M}$ , there exists a unique $\xi \in S^{[3]}$ such that $h^1(E\otimes \mathcal {I}_\xi )=2$ ; it is nothing but the image under $\gamma $ of the unique noncurvilinear length 3 subscheme supported at $E\in \mathcal {M}$ . This concludes the proof of the proposition.

Proof. The arguments are similar to those of the previous subsection. For instance, in (1) one sends $M\in \mathcal {M}(2,-L,5)$ to the isomorphism class of the object P sitting in a nontrivial extension $0\to M\to P\to \overline {G}[1]\to 0$ .

Proof of Proposition 5.6.

Given $\xi \in S^{[4]}$ and $E'\in \mathcal {M}$ , we want to determine the number $\hom (E^{\prime \vee },\mathcal {I}_\xi )=h^0(E'\otimes \mathcal {I}_\xi )$ . We will use the isomorphism $\overline {\gamma }$ of the previous lemma, together with the fact that $E^{\prime \vee }\in \mathcal {M}(2,-L,5)$ . Consider the HN filtration

$$\begin{align*}0\to \overline{F}\to \mathcal{I}_\xi\to \overline{G}[1]\to 0 \end{align*}$$

of $\mathcal {I}_\xi $ for stability conditions just below $\overline {\sigma _0}$ ; the factor $\overline {F}$ is determined by $\overline {\gamma }^{-1}(\xi )$ . Indeed, one has $\overline {F}=\Phi _{\overline {\mathcal {E}}}(\mathcal {O}_{\overline {\gamma }^{-1}(\xi )})$ , where $\overline {\mathcal {E}}$ denotes the universal bundle on $S\times \mathcal {M}(2,-L,5)$ and $\Phi _{\overline {\mathcal {E}}}:\mathop {\mathrm {D}^{\mathrm {b}}}\nolimits (\mathcal {M}(2,-L,5))\to \mathop {\mathrm {D}^{\mathrm {b}}}\nolimits (S)$ is the corresponding Fourier–Mukai equivalence.

Due to the vanishing $\operatorname {\mathrm {Hom}}(E^{\prime \vee },\overline {G}[1])=0$ , we have

$$\begin{align*}\begin{aligned} \operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)}(E^{\prime\vee},\mathcal{I}_\xi)&=\operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)}(E^{\prime\vee}, \overline{F})=\operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)}(\Phi_{\overline{\mathcal{E}}}(k(E^{\prime\vee}), \Phi_{\overline{\mathcal{E}}}(\mathcal{O}_{\overline{\gamma}^{-1}(\xi)}))=\\ &=\operatorname{\mathrm{Hom}}_{\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathcal{M}(2,-L,5))}(k(E^{\prime\vee}),\mathcal{O}_{\overline{\gamma}^{-1}(\xi)}). \end{aligned} \end{align*}$$

If $E^{\prime \vee }\notin \operatorname {\mathrm {supp}} \overline {\gamma }^{-1}(\xi )$ , the vanishing $\hom (E^{\prime \vee },\mathcal {I}_\xi )=0$ follows.

On the other hand, if $E^{\prime \vee }\in \operatorname {\mathrm {supp}} \overline {\gamma }^{-1}(\xi )$ , we have the equality

$$\begin{align*}\hom(k(E^{\prime\vee}),\mathcal{O}_{\overline{\gamma}^{-1}(\xi)})=\operatorname{\mathrm{ext}}^1(k(E^{\prime\vee}),\mathcal{O}_{\overline{\gamma}^{-1}(\xi)})-1=\operatorname{\mathrm{ext}}^1(\mathcal{O}_{\overline{\gamma}^{-1}(\xi)},k(E^{\prime\vee}))-1 \end{align*}$$

(following from $\operatorname {\mathrm {ext}}^2(k(E^{\prime \vee }),\mathcal {O}_{\overline {\gamma }^{-1}(\xi )})=\hom (\mathcal {O}_{\overline {\gamma }^{-1}(\xi )},k(E^{\prime \vee }))=1$ and $\chi (k(E^{\prime \vee }), \mathcal {O}_{\overline {\gamma }^{-1}(\xi )})=0$ ). Therefore, by applying Lemma 2.6 we deduce that

$$\begin{align*}\hom(E^{\prime\vee}, \mathcal{I}_\xi)=\left\{\!\begin{array}{c l} 1 & \text{if }\overline{\gamma}^{-1}(\xi)\text{ is of complete intersection at }E^{\prime\vee}\\ 2 & \text{if }\overline{\gamma}^{-1}(\xi)\text{ is not of complete intersection at }E^{\prime\vee}\\ \end{array} \right. \end{align*}$$

which finishes the proof.

Proof. If $\xi $ consists of a reduced point $p\in S$ , then by Lemma 2.4 there is a local trivialization $E_p\cong e_1\cdot \mathcal {O}_{S,p}\oplus e_2\cdot \mathcal {O}_{S,p}$ of E near p such that $s_p\in m_pe_1+m_p^4 e_2$ for every $s\in V^\vee $ . This implies $\mathcal {I}\subset \mathcal {I}_p^5$ and hence $\operatorname {\mathrm {colength}}(\mathcal {I})\geq \operatorname {\mathrm {colength}}(\mathcal {I}_p^5)=15$ , which is a contradiction.

Assume now that $\xi $ is of length 2, supported at two distinct points p and q. If $\bigcap _{s\in V^\vee }Z_{cycle}(s)=2p+2q$ , then by Lemma 2.4 one has $s_p\in m_pe_1+m_p^2e_2$ for all $s\in V^\vee $ , in an appropriate trivialization of E near p; therefore, $\mathcal {I}\subset \mathcal {I}_p^3$ , and since $\mathcal {I}$ has (at most) three local generators at p, it follows that $\operatorname {\mathrm {colength}}(\mathcal {I},p)\geq 7$ . Similarly, we have $\operatorname {\mathrm {colength}}(\mathcal {I},q)\geq 7$ , hence $\operatorname {\mathrm {colength}}(\mathcal {I})\geq 14$ which is impossible.

On the other hand, if $\bigcap _{s\in V^\vee }Z_{cycle}(s)=3p+q$ , a similar use of Lemma 2.4 yields $\mathcal {I}\subset \mathcal {I}_p^4\cap \mathcal {I}_q^2$ ; since $\mathcal {I}$ has three generators at p, we have $\operatorname {\mathrm {colength}}(\mathcal {I},p)\geq 12$ and therefore $\operatorname {\mathrm {colength}}(\mathcal {I})\geq \operatorname {\mathrm {colength}}(\mathcal {I},p)+\operatorname {\mathrm {colength}}(\mathcal {I},q)\geq 15$ , contradiction.

It only remains to discard the situation where $\xi $ is nonreduced of length 2, supported at a point p. In this case, by Lemma 2.4 we can find (in appropriate local coordinates) generators $e_1,e_2$ of E around p such that $s_p\in (x,y^2)e_1+(x^2,xy^2,y^4)e_2$ for every $s\in V^\vee $ . Under the isomorphism $V\cong \bigwedge ^2V^\vee $ , this implies that $\mathcal {I}\subset (x^3,x^2y^2,xy^4,y^6)$ locally at p; since $\mathcal {I}$ has at most three generators at p, it follows that $\operatorname {\mathrm {colength}}(\mathcal {I},p)\geq 14$ , which gives a contradiction.

Proof. Let us first show that $\xi $ is curvilinear. Assume for the sake of a contradiction that $\xi $ is not curvilinear, namely $\mathcal {I}_\xi =\mathcal {I}_p^2$ . Since $h^0(E\otimes \mathcal {I}_\xi )\geq 3$ , we have $h^1(E\otimes \mathcal {I}_\xi )\geq 2$ and we can consider a stable sheaf F arising as a double extension

$$\begin{align*}0\longrightarrow\mathcal{O}_S^{\oplus 2}\longrightarrow F\longrightarrow E\otimes\mathcal{I}_p^2\longrightarrow 0. \end{align*}$$

Note that $v(F)=(4,L,1)$ . Since $E\otimes \mathcal {I}_p^2\otimes k(p)$ is a six-dimensional ( $E\otimes \mathcal {I}_p^2$ has six local generators at p), it follows that the minimal number of generators of F at p is at least six. This implies that the reflexive hull $F^{**}$ of F has Mukai vector $v(F^{**})=(4,L,k)$ with $k\geq 3$ , hence $v(F^{**})^2\leq -4$ which contradicts the stability of $F^{**}$ .

Now, we only need to rule out the scenario where $\xi $ is of the form $(x,y^m)$ locally at p for $m\in \{2,3\}$ . In that case, by Lemma 2.4 we have $s_p\in (x,y^m)\cdot e_1+(x^2,xy,y^{m+1})\cdot e_2$ for all $s\in V^\vee $ in an appropriate trivialization $E_p\cong e_1\cdot \mathcal {O}_{S,p}\oplus e_2\cdot \mathcal {O}_{S,p}$ . In view of the isomorphism $V\cong \bigwedge ^2V^\vee $ , it follows that the base ideal $\mathcal {I}$ satisfies $\mathcal {I}\otimes \hat {\mathcal {O}}_{S,p}\subset (x^3,x^2y, xy^{m+1},y^{2m+1})$ locally at p; furthermore, recall that $\mathcal {I}$ has at most three local generators at p.

For $m=3$ , this implies that $\operatorname {\mathrm {colength}}(\mathcal {I},p)\geq 14$ , which is a contradiction.

For $m=2$ , this implies that $\operatorname {\mathrm {colength}}(\mathcal {I},p)\geq 11$ . If $q\neq p$ is the other point where $\xi $ is supported, then $\operatorname {\mathrm {colength}}(\mathcal {I},q)\geq 3$ ; hence, $\operatorname {\mathrm {colength}}(\mathcal {I})\geq 14$ , and again we obtain a contradiction.

Proof. Let us denote by $\zeta $ the (length 2) residual subscheme of p in $\xi $ . Let $\mathcal {E}$ be a universal bundle on $S\times \mathcal {M}$ , defining a Fourier–Mukai equivalence $\widetilde {\Phi _{\mathcal {E}}}:=\Phi _{\mathcal {E}}^{S\to \mathcal {M}}:\mathop {\mathrm {D}^{\mathrm {b}}}\nolimits (S)\to \mathop {\mathrm {D}^{\mathrm {b}}}\nolimits (\mathcal {M})$ .

The first observation is that, for an appropriate choice of $\mathcal {E}$ (obtained by twisting $\mathcal {E}$ with the pullback of a line bundle on $\mathcal {M}$ ), we have $R^0\widetilde {\Phi _{\mathcal {E}}}(\mathcal {I}_\xi )\cong \mathcal {O}_{\mathcal {M}}$ . This can be deduced from the following facts:

• • $v(\widetilde {\Phi _{\mathcal {E}}}(\mathcal {I}_\xi ))=(1,0,-2)$ (see, e.g., [Reference YoshiokaYos01, Lemma 7.2] or [Reference MaciociaMac17, Proposition 15]).

• • $R^1\widetilde {\Phi _{\mathcal {E}}}(\mathcal {I}_\xi )={\mathcal {E}\mathit {xt}}^2(\mathcal {O}_{\gamma ^{-1}(\xi )},\mathcal {O}_{\mathcal {M}})$ , where $\gamma :\mathcal {M}^{[3]}\to S^{[3]}$ is the isomorphism of Lemma 5.5. Indeed, to prove this equality one simply applies the Fourier-Mukai functor $\widetilde {\Phi _{\mathcal {E}}}$ to the distinguished triangle

  $$\begin{align*}0\to G\to \mathcal{I}_\xi\to \Phi^{\mathcal{M}\to S}_{\mathcal{E}}(\mathcal{O}_{\gamma^{-1}(\xi)})^\vee[1]\to 0 \end{align*}$$
  (see the proof of Proposition 5.1), and uses that $\Phi ^{\mathcal {M}\to S}_{\mathcal {E}}(\cdot ^\vee )^\vee $ is the quasi-inverse of $\widetilde {\Phi _{\mathcal {E}}}$ .

The unique (up to constant) global section of $R^0\widetilde {\Phi _{\mathcal {E}}}(\mathcal {I}_\xi )\cong \mathcal {O}_{\mathcal {M}}$ restricts, at a point $E'\in \mathcal {M}\setminus \gamma ^{-1}(\xi )$ , to the unique section of $H^0(S,E'\otimes \mathcal {I}_\xi )$ . We will denote by $\widetilde {s}\in H^0(S\times \mathcal {M},\pi _1^*(\mathcal {I}_\xi )\otimes \mathcal {E})$ the corresponding section, where $\pi _1:S\times \mathcal {M}\to S$ is the first projection.

Locally, at a point $E'\in \mathcal {M}$ , the locus $C_{p,\xi }$ is given as follows. Fix $x,y$ local analytic coordinates of S around p, $z,w$ local coordinates of $\mathcal {M}$ around $E'$ and a local trivialization of $\mathcal {E}$ around $(p,E')\in S\times \mathcal {M}$ . Then $\tilde s$ has a local expression

$$\begin{align*}(a_1(z,w)\cdot x+b_1(z,w)\cdot y+O(x,y)^2,a_2(z,w)\cdot x+b_2(z,w)\cdot y+O(x,y)^2) \end{align*}$$

and $C_{p,\xi }$ is locally given by the equation $a_1b_2-a_2b_1=0$ .

To determine the class of the curve $C_{p,\xi }$ , we pick a basis $w_1,w_2$ of the tangent space $T_pS$ at p. For $i=1,2$ , the section $\tilde s$ gives rise to a section

$$\begin{align*}\tilde{s_i}\in H^0(S\times\mathcal{M},\pi_1^*\left(\mathcal{I}_{\xi}/(\mathcal{I}_\zeta\otimes\mathcal{I}_{[w_i]})\right)\otimes\mathcal{E})=H^0(\mathcal{M},R^0\widetilde{\Phi_{\mathcal{E}}}\left(\mathcal{I}_{\xi}/(\mathcal{I}_\zeta\otimes\mathcal{I}_{[w_i]})\right)). \end{align*}$$

Now, observe that $\mathcal {I}_{\xi }/(\mathcal {I}_\zeta \otimes \mathcal {I}_{[w_i]})$ is isomorphic (as an $\mathcal {O}_S$ -module) to the skyscraper sheaf $k(p)$ ; this isomorphism is canonical (up to constant), defined via $w_i$ . We obtain an isomorphism

$$\begin{align*}R^0\widetilde{\Phi_{\mathcal{E}}}\left(\mathcal{I}_{\xi}/(\mathcal{I}_\zeta\otimes\mathcal{I}_{[w_i]})\right)\cong R^0\widetilde{\Phi_{\mathcal{E}}}\left(k(p)\right)=\widetilde{\Phi_{\mathcal{E}}}(k(p))=\mathcal{E}_p \end{align*}$$

given by $w_i$ , where $\mathcal {E}_p\cong \mathcal {E}|_{\{p\}\times \mathcal {M}}$ is the vector bundle on $\mathcal {M}$ defined by the point $p\in S$ .

The curve $C_{p,\xi }$ is then defined as the zero locus of $\tilde {s_1}\wedge \tilde {s_2}\in H^0(\bigwedge ^2 \mathcal {E}_p)$ . Since $v(\mathcal {E}_p)=(2,\hat {L},5)$ (see again [Reference MaciociaMac17, Proposition 15])), we have $\bigwedge ^2 \mathcal {E}_p=\hat {L}$ and the assertion follows.

Proof. We argue by contradiction. As in the proof of Lemma 5.12, we denote by $\zeta $ the (length 2) residual subscheme of p in $\bigcap _{s\in V^\vee }Z(s)$ ; in the notations of the statement, we have $\zeta =q+r$ with $q,r$ two (possibly infinitely near) points.

First, notice that for any length 2 subscheme $\eta \in \mathbb {P}(T_pS)$ nonreduced at p, the inequality $h^0(E\otimes \mathcal {I}_{\eta \cup \zeta })\ge 2$ holds; this follows from Lemma 2.4. Hence, the subschemes $\{\eta \cup \zeta \;|\;\eta \in \mathbb {P}(T_pS)\}$ describe a smooth rational curve in the jump locus $\mathrm {Bl}_{E}(\mathcal {M})\subset S^{[4]}$ of Corollary 5.9. We claim that this curve is the exceptional divisor in $\mathrm {Bl}_{E}(\mathcal {M})$ ; in particular, by Corollary 5.9.(2) this will imply that $h^0(\tilde E \otimes \mathcal {I}_{\eta \cup \zeta })=0$ for any $\eta \in \mathbb {P}(T_pS)$ and $\tilde E \in \mathcal {M}\setminus~\{ E\}$ .

In order to prove the claim, assume that the curve is not the exceptional divisor. Then its image R in $\mathcal {M}$ is a rational curve, describing a component of the curve

$$\begin{align*}C_{p,\xi}:=\{E'\in\mathcal{M}\;|\;\exists s\in H^0(E'\otimes\mathcal{I}_\xi)\setminus\{0\}\text{ such that }Z(s)\text{ has length }\geq 2\text{ at }p\}. \end{align*}$$

The curve $C_{p,\xi }$ lies in the primitive linear system $|\hat {L}|$ of $\mathcal {M}$ by Lemma 5.12, hence $R=C_{p,\xi }\in |\hat {L}|$ and $p_a(R)=11$ . This yields a contradiction since on the one hand R can only be singular at $E\in \mathcal {M}$ , whereas on the other hand every rational curve in $|\hat {L}|$ is nodal under our generality assumptions on $(S,L)$ (hence on $(\mathcal {M},\hat {L})$ ) by Chen’s result [Reference ChenChe02].

Now, we start the analysis. Under our hypotheses, we get $V=\bigwedge ^2 V^\vee \subset H^0(L\otimes \mathcal {I}_p^3\otimes \mathcal {I}_\zeta ^2)$ . Given $z,w$ local coordinates at p, consider for every $\lambda \in \mathbb {C}$ the colength 4 ideal $\mathcal {J}_\lambda :=(z(z+\lambda w),w(z+\lambda w),w^3)\supset \mathcal {I}_p^3$ . Since $V\subset H^0(L \otimes \mathcal J_\lambda \otimes \mathcal {I}_\zeta ^2)$ , we get $h^1(L \otimes \mathcal J_\lambda \otimes \mathcal {I}_\zeta ^2)\ge 1$ .