Now we prove that (iii) implies (i). Suppose that $\overline{A}' - \overline{A} \in \mathfrak{m}\mathcal{I}$ and $\overline{B}' - \overline{B} \in \mathfrak{m}\mathcal{I}$. So we can write $\overline{A}' = \overline{A} + x_0\overline{u} + y_0\overline{v}$, with $\overline{u}, \overline{v} \in \mathcal{I}$. Hence

$$x + \overline{A}' = x + \overline{A} + x_0\overline{u} + y_0\overline{v} = (1 + \overline{u})(x + \overline{A}) + (y + \overline{B})\overline{v},$$

(since $\overline{A}\overline{u} = \overline{B}\overline{v} = 0$) so $x + \overline{A}' \in (x + \overline{A}, y + \overline{B})$. Similarly $y + \overline{B}' \in (x + \overline{A}, y + \overline{B})$. Hence $(x + \overline{A}', y + \overline{B}') \subset (x + \overline{A}, y + \overline{B})$. In the same way $(x + \overline{A}, y + \overline{B}) \subset (x + \overline{A}', y + \overline{B}')$ and (i) is true.

Suppose now that (iii) is true. Then we have proved that (i) is true, and so is (iv). Conversely, suppose that (iv) is true. Let $\sigma : (x + \overline{A}, y + \overline{B}) \to (x + \overline{A}', y + \overline{B}')$ be an isomorphism. It induces an automorphism of the $\mathcal{O}_1$-module $(x_0, y_0)$, which is the multiplication by some invertible element of $\mathcal{O}_1$. Hence with respect to the generators $x + \overline{A}$, $y + \overline{B}$ of $(x + \overline{A}, y + \overline{B})$, and $x + \overline{A}'$, $y + \overline{B}'$ of $(x + \overline{A}', y + \overline{B}')$, $\sigma$ is represented by a matrix $\begin{pmatrix} \alpha + \overline{u} & \overline{v} \\ \overline{w} & \alpha + \overline{z} \end{pmatrix}$, with $\alpha \in \mathcal{O}_2$ invertible, $\overline{u}, \overline{v}, \overline{w}, \overline{z} \in \mathcal{I}$. Replacing $\sigma$ with $\alpha^{-1}\sigma$, we can assume that $\alpha = 1$. The equation

$$(y + \overline{B}')\sigma(x + \overline{A}') = (x + \overline{A}')\sigma(y + \overline{B}')$$

gives

$$x_0[y_0(\overline{u} - \overline{z}) + \overline{B}' - \overline{B} + x_0\overline{w}] = y_0[y_0\overline{v} + \overline{A}' - \overline{A}]$$

in $\mathcal{I}$. It follows that $\overline{A}' - \overline{A} \in \mathfrak{m}\mathcal{I}$, and similarly $\overline{B}' - \overline{B} \in \mathfrak{m}\mathcal{I}$. So (iii) is true. $\square$

## 7.2. EXTENSIONS OF IDEALS OF $\mathcal{O}_{X,P}$

**7.2.1. Theorem:** The balanced $\mathcal{O}_2$-modules $M$ such that $M \otimes_{\mathcal{O}_2} \mathcal{O}_1 \simeq \mathfrak{m}$ are the ideals of $\mathcal{J}_P$.

*Proof.* The ideals of $\mathcal{J}_P$ are balanced modules by proposition 3.7.2, and if $\mathcal{J} \in \mathcal{J}_P$, then $\mathcal{J} \otimes_{\mathcal{O}_2} \mathcal{O}_1 \simeq \mathfrak{m}$. Conversely we will study the extensions of $\mathcal{O}_2$-modules

$$0 \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \to M \to \mathfrak{m} \to 0,$$

and prove that when $M$ is balanced, then it is an ideal of $\mathcal{J}_P$.

We have the following free resolution of $\mathfrak{m}$ as a $\mathcal{O}_2$-module:

$$(7) \qquad 3\mathcal{O}_2 \xrightarrow{\phi_2} 3\mathcal{O}_2 \xrightarrow{\phi_1} 2\mathcal{O}_2 \xrightarrow{\phi_0} \mathfrak{m} \longrightarrow 0,$$

where $\phi_0, \phi_1, \phi_2$ are represented by the matrices

$$(x \ y) , \quad \begin{pmatrix} y & \bar{t} & 0 \\ -x & 0 & \bar{t} \end{pmatrix} , \quad \begin{pmatrix} \bar{t} & 0 & 0 \\ -y & \bar{t} & 0 \\ x & 0 & \bar{t} \end{pmatrix}$$

respectively. We have

$$K = \ker(\phi_0) = \{(ey + \bar{\gamma}, -ex + \bar{\delta}) ; e \in \mathcal{O}_2, \bar{\gamma}, \bar{\delta} \in \mathcal{J}\}.$$

**Step 1.** Morphisms from $K$ to $\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} = \mathfrak{m}\mathcal{I}$ – Recall that $\mathcal{J}$ is isomorphic to $\mathcal{O}_1$, so $\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \simeq \mathfrak{m}$. These morphisms are the $\Theta_{\bar{\tau},\rho}, \bar{\tau} \in \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}, \rho \in \mathcal{O}_1$, with

$$\Theta_{\bar{\tau},\rho}(ey + \bar{\gamma}, -ex + \bar{\delta}) = e_0\bar{\tau} + \rho(x_0\bar{\gamma} + y_0\bar{\delta}).$$
Now we prove that (i) implies (ii). Suppose that (i) is true. There exist $\lambda, \mu, \epsilon, \rho \in \mathcal{O}_2$ such that

$$ (6) \qquad x + \bar{A}' = \lambda(x + \bar{A}) + \mu(y + \bar{B}), \quad y + \bar{B}' = \epsilon(x + \bar{A}) + \rho(y + \bar{B}). $$

We have $\lambda_0 x_0 + \mu_0 y_0 = x_0$, hence there is some $\Psi \in \mathcal{O}_2$ such that $\lambda_0 = 1 - \Psi y_0$ and $\mu_0 = \Psi x_0$. Similarly there exists $\theta \in \mathcal{O}_2$ such that $\epsilon_0 = \theta y_0$ and $\rho_0 = 1 - \theta x_0$. Hence we can write

$$ \lambda = 1 - \psi y + \bar{w}, \quad \mu = \psi x + \bar{z}, \quad \epsilon = \theta y + \bar{u}, \quad \rho = 1 - \theta x + \bar{v}, $$

with $\bar{w}, \bar{z}, \bar{u}, \bar{v} \in \mathcal{J}$. It follows that, with $\bar{\tau} = -y_0 \bar{A} + x_0 \bar{B}, \bar{\tau}' = -y_0 \bar{A}' + x_0 \bar{B}'$,

$$ \bar{\tau}' - \bar{\tau} = -x_0 y_0 \bar{w} - y_0^2 \bar{z} - y_0 \psi_0 \bar{\tau} + x_0^2 \bar{u} + x_0 y_0 \bar{v} - x_0 \theta_0 \bar{\tau}, $$

hence $\tau = \tau'$ and (ii) is true.

Now we prove that (iii) implies (i). Suppose that $\bar{A}' - \bar{A} \in \mathfrak{m}\mathcal{J}$ and $\bar{B}' - \bar{B} \in \mathfrak{m}\mathcal{J}$. So we can write $\bar{A}' = \bar{A} + x_0 \bar{u} + y_0 \bar{v}$, with $\bar{u}, \bar{v} \in \mathcal{J}$. Hence

$$ x + \bar{A}' = x + \bar{A} + x_0 \bar{u} + y_0 \bar{v} = (1 + \bar{u})(x + \bar{A}) + (y + \bar{B})\bar{v}, $$

(since $\bar{A}\bar{u} = \bar{B}\bar{v} = 0$) so $x + \bar{A}' \in (x + \bar{A}, y + \bar{B})$. Similarly $y + \bar{B}' \in (x + \bar{A}, y + \bar{B})$. Hence $(x + \bar{A}', y + \bar{B}') \subset (x + \bar{A}, y + \bar{B})$. In the same way $(x + \bar{A}, y + \bar{B}) \subset (x + \bar{A}', y + \bar{B}')$ and (i) is true.

Suppose now that (iii) is true. Then we have proved that (i) is true, and so is (iv). Conversely, suppose that (iv) is true. Let $\sigma : (x + \bar{A}, y + \bar{B}) \to (x + \bar{A}', y + \bar{B}')$ be an isomorphism. It induces an automorphism of the $\mathcal{O}_1$-module $(x_0, y_0)$, which is the multiplication by some invertible element of $\mathcal{O}_1$. Hence with respect to the generators $x + \bar{A}, y + \bar{B}$ of $(x + \bar{A}, y + \bar{B})$, and $x + \bar{A}', y + \bar{B}'$ of $(x + \bar{A}', y + \bar{B}')$, $\sigma$ is represented by a matrix $\begin{pmatrix} \alpha + \bar{u} & \bar{v} \\ \bar{w} & \alpha + \bar{z} \end{pmatrix}$, with $\alpha \in \mathcal{O}_2$ invertible, $\bar{u}, \bar{v}, \bar{w}, \bar{z} \in \mathcal{J}$. Replacing $\sigma$ with $\alpha^{-1}\sigma$, we can assume that $\alpha = 1$. The equation

$$ (y + \bar{B}')\sigma(x + \bar{A}') = (x + \bar{A}')\sigma(y + \bar{B}') $$

gives

$$ x_0[y_0(\bar{u} - \bar{z}) + \bar{B}' - \bar{B} + x_0\bar{w}] = y_0[y_0\bar{v} + \bar{A}' - \bar{A}] $$

in $\mathcal{J}$. It follows that $\bar{A}' - \bar{A} \in \mathfrak{m}\mathcal{J}$, and similarly $\bar{B}' - \bar{B} \in \mathfrak{m}\mathcal{J}$. So (iii) is true. $\square$

## 7.2. EXTENSIONS OF IDEALS OF $\mathcal{O}_{X,P}$

**7.2.1. Theorem:** The balanced $\mathcal{O}_2$-modules $M$ such that $M \otimes_{\mathcal{O}_2} \mathcal{O}_1 \simeq \mathfrak{m}$ are the ideals of $\mathcal{J}_P$.

*Proof.* The ideals of $\mathcal{J}_P$ are balanced modules by proposition 3.7.2, and if $\mathcal{J} \in \mathcal{J}_P$, then $\mathcal{J} \otimes_{\mathcal{O}_2} \mathcal{O}_1 \simeq \mathfrak{m}$. Conversely we will study the extensions of $\mathcal{O}_2$-modules

$$ 0 \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \to M \to \mathfrak{m} \to 0, $$

and prove that when $M$ is balanced, then it is an ideal of $\mathcal{J}_P$.

We have the following free resolution of $\mathfrak{m}$ as a $\mathcal{O}_2$-module:

$$ (7) \qquad 3\mathcal{O}_2 \xrightarrow{\phi_2} 3\mathcal{O}_2 \xrightarrow{\phi_1} 2\mathcal{O}_2 \xrightarrow{\phi_0} \mathfrak{m} \longrightarrow 0, $$
**Step 2.** *Description of Ext<sub>0</sub><sup>1</sup>(m, m ⊗<sub>0</sub><sub>1</sub> J) – From (7) follows the complex*

$$
\begin{tikzcd}
& & 2(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \arrow[r, "\Psi_1"] & 3(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \arrow[r, "\Psi_2"] & 3(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \\
& \operatorname{Hom}(2\mathcal{O}_2, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) & \operatorname{Hom}(3\mathcal{O}_2, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) & \operatorname{Hom}(3\mathcal{O}_2, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}), & \operatorname{Hom}(3\mathcal{O}_2, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}),
\end{tikzcd}
$$

(whence $\operatorname{Ext}_{\mathcal{O}_2}^1(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \simeq \ker(\Psi_2)/\operatorname{im}(\Psi_1)$) where $\Psi_1, \Psi_2$ are represented by the matrices

$$
\begin{pmatrix} y_0 & -x_0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & y_0 & -x_0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}
$$

respectively. It follows easily that we have

$$
\ker(\Psi_2) = \{(\alpha, x_0 \otimes \bar{\epsilon}, y_0 \otimes \bar{\epsilon}); \alpha \in \mathfrak{m}, \bar{\epsilon} \in \mathcal{J}\} = (\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \oplus W,
$$

with $W = (x_0, y_0) \otimes \mathcal{J} \subset 2(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})$,

$$
\mathrm{im}(\Psi_1) = \mathfrak{m}^2 \otimes_{\mathcal{O}_1} \mathcal{J} \subset \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \quad (\text{the first factor in } 3(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})) .
$$

$$
\text{Hence } \operatorname{Ext}_{\mathcal{O}_2}^{1}(\mathfrak{m}, \mathfrak{m} \otimes \mathcal{J}_X) \simeq ((\mathfrak{m}/\mathfrak{m}^2) \otimes_{\mathcal{O}_1} \mathcal{J}) \oplus \mathcal{O}_1 .
$$

**Step 3.** *Description of the extensions* – The morphisms $f : 3\mathcal{O}_2 \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$ such that $f(\mathrm{im}(\phi_2)) = \{0\}$ (i.e. the elements of $\ker(\Psi_2)$) are the $f_{\bar{\tau},\rho}$, with $\bar{\tau} \in \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$, $\rho \in \mathcal{O}_1$ and

$$
f_{\bar{\tau}, \rho}(a, b, c) = a_0 \bar{\tau} + \rho(x_0 b_0 \bar{t} + y_0 c_0 \bar{t}),
$$

inducing $\Theta_{\bar{\tau},\rho} : K \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$. Let $\tau \in (\mathfrak{m}/\mathfrak{m}^2) \otimes_{\mathcal{O}_1} \mathcal{J}$ be the image of $\bar{\tau}$. Then the element of $\mathrm{Ext}^1_{\mathcal{O}_2}(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})$ corresponding to $f_{\bar{\tau},\rho}$ is $(\tau, \rho)$.

Let

$$
(8) \qquad 0 \longrightarrow \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \xrightarrow{i} M \xrightarrow{p} \mathfrak{m} \longrightarrow 0
$$

be the extension associated to the image of $f_{\bar{\tau},\rho}$ in $\operatorname{Ext}_{\mathcal{O}_2}^1(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})$. We have an exact sequence

$$
(9) \quad 0 \longrightarrow K \xrightarrow{g=(\iota, \Theta_{\bar{\tau}, \rho})} 2\mathcal{O}_2 \oplus (\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \xrightarrow{q} M \longrightarrow 0,
$$

where $\iota$ is the inclusion. Hence we have

$$
(10) \quad \mathrm{im}(g) = \{ (ey + \bar{\gamma}, -ex + \bar{\delta}, e_0\bar{\tau} + \rho(x_0\bar{\gamma} + y_0\bar{\delta})); e \in \mathcal{O}_2, \bar{\gamma}, \bar{\delta} \in \mathcal{J} \} .
$$

In the exact sequence 8, $i: \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \to M$ comes from the inclusion

$$
\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \hookrightarrow 2\mathcal{O}_2 \oplus (\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}), \text{ and } q: M \to \mathfrak{m} \text{ from } \phi_0.
$$

It follows easily from (9) that $M$ is a $\mathcal{O}_1$-module (i.e. $\mathcal{J}M = \{0\}$) if and only if $\rho = 0$.

**Step 4.** *Properties of the extensions* – We will describe the morphism $h : \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$ induced by the canonical morphism $M \otimes_{\mathcal{O}_2} \mathcal{J} \to M$ and (8) ($M$ is balanced if and only if $h$ is an isomorphism). Let $\alpha \in \mathfrak{m}$, that we can write as $\alpha = \phi_0(a, b) = a_0x_0 + b_0y_0$, with $a, b \in \mathcal{O}_2$. Then from step 3, $h(\alpha \otimes \bar{t})$ is the unique $\beta \otimes \bar{t} \in \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$ such that $(0, 0, \beta \otimes \bar{t}) - (\bar{t}a, \bar{t}b, 0) \in g(K)$. Using (9), we see that $\beta = -\rho\alpha$, so that $h$ is the multiplication by $-\rho$.
where $\phi_0, \phi_1, \phi_2$ are represented by the matrices

$$ (x \ y) , \quad \begin{pmatrix} y & \bar{t} & 0 \\ -x & 0 & \bar{t} \end{pmatrix} , \quad \begin{pmatrix} \bar{t} & 0 & 0 \\ -y & \bar{t} & 0 \\ x & 0 & \bar{t} \end{pmatrix} $$

respectively. We have

$$ K = \ker(\phi_0) = \{(ey + \bar{\gamma}, -ex + \bar{\delta}) ; e \in \mathcal{O}_2, \bar{\gamma}, \bar{\delta} \in \mathcal{J}\}. $$

**Step 1.** *Morphisms from $K$ to $\mathfrak{m} \otimes_{\mathcal{O}_2} \mathcal{J} = \mathfrak{m}\mathcal{J}$ – Recall that $\mathcal{J}$ is isomorphic to $\mathcal{O}_1$, so $\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \simeq \mathfrak{m}$. These morphisms are the $\Theta_{\bar{\tau},\rho}, \bar{\tau} \in \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}, \rho \in \mathcal{O}_1$, with*

$$ \Theta_{\bar{\tau},\rho}(ey + \bar{\gamma}, -ex + \bar{\delta}) = e_0\bar{\tau} + \rho(x_0\bar{\gamma} + y_0\bar{\delta}). $$

**Step 2.** *Description of $\operatorname{Ext}^1_{\mathcal{O}_2}(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})$ – From (7) follows the complex*

$$ \begin{tikzcd} 2(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \arrow[r, "\Psi_1"] & 3(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \arrow[r, "\Psi_2"] & 3(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \arrow[d, "\cong"] \\ \operatorname{Hom}(2\mathcal{O}_2, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) & \operatorname{Hom}(3\mathcal{O}_2, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) & \operatorname{Hom}(3\mathcal{O}_2, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}), \end{tikzcd} $$

(whence $\operatorname{Ext}^1_{\mathcal{O}_2}(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \simeq \ker(\Psi_2)/\operatorname{im}(\Psi_1)$) where $\Psi_1, \Psi_2$ are represented by the matrices

$$ \begin{pmatrix} y_0 & -x_0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & y_0 & -x_0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} $$

respectively. It follows easily that we have

$$ \ker(\Psi_2) = \{(\alpha, x_0 \otimes \bar{\epsilon}, y_0 \otimes \bar{\epsilon}) ; \alpha \in \mathfrak{m}, \bar{\epsilon} \in \mathcal{J}\} = (\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \oplus W, $$

with $W = (x_0, y_0) \otimes \mathcal{J} \subset 2(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})$,

$$ \operatorname{im}(\Psi_1) = \mathfrak{m}^2 \otimes_{\mathcal{O}_1} \mathcal{J} \subset \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \quad (\text{the first factor in } 3(\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})) . $$

$$ \text{Hence } \operatorname{Ext}^1_{\mathcal{O}_2}(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{J}_X}) \simeq ((\mathfrak{m}/\mathfrak{m}^2) \otimes_{\mathcal{O}_1} \mathcal{J}) \oplus \mathcal{O}_1 . $$

**Step 3.** *Description of the extensions* – The morphisms $f : 3\mathcal{O}_2 \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$ such that $f(\operatorname{im}(\phi_2)) = \{0\}$ (i.e. the elements of $\ker(\Psi_2)$) are the $f_{\bar{\tau},\rho}$, with $\bar{\tau} \in \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}, \rho \in \mathcal{O}_1$ and

$$ f_{\bar{\tau},\rho}(a, b, c) = a_0\bar{\tau} + \rho(x_0b_0\bar{t} + y_0c_0\bar{t}), $$

inducing $\Theta_{\bar{\tau},\rho} : K \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$. Let $\tau \in (\mathfrak{m}/\mathfrak{m}^2) \otimes_{\mathcal{O}_1} \mathcal{J}$ be the image of $\bar{\tau}$. Then the element of $\operatorname{Ext}^1_{\mathcal{O}_2}(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{J}_X})$ corresponding to $f_{\bar{\tau},\rho}$ is $(\tau, \rho)$.

Let

$$ (8) \qquad 0 \longrightarrow \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \xrightarrow{i} M \xrightarrow{p} \mathfrak{m} \longrightarrow 0 $$

be the extension associated to the image of $f_{\bar{\tau},\rho}$ in $\operatorname{Ext}^1_{\mathcal{O}_2}(\mathfrak{m}, \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J})$. We have an exact sequence

$$ (9) \qquad 0 \longrightarrow K \xrightarrow{g=(\iota, \Theta_{\bar{\tau},\rho})} 2\mathcal{O}_2 \oplus (\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}) \xrightarrow{q} M \longrightarrow 0, $$
Now suppose that $\rho \neq 0$. Then it is easy to see that $M^{(1)} = \text{im}(i) = \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{I}$, and that $M^{(1)}/M_1 \simeq \mathfrak{m}/\rho\mathfrak{m}$. It follows that $M$ is balanced if and only if $\rho$ is invertible, and that (8) is the canonical exact sequence $0 \to M^{(1)} \to M \to M^{(0)} \to 0$ if and only if $\rho = -1$.

**Step 5.** *Description of the morphisms $M \to \mathcal{O}_2$* – We suppose that $\rho = -1$. Such a morphism comes from $\psi : 2\mathcal{O}_2 \oplus \mathfrak{m}\mathcal{I} \to \mathcal{O}_2$ such that $\psi \circ g = 0$. There exist $a, b \in \mathcal{O}_2$, $\alpha \in \mathcal{O}_1$, such that $\psi(u, v, \bar{w}) = au + bv + \alpha\bar{w}$ for every $u, v \in \mathcal{O}_2$, $\bar{w} \in \mathfrak{m}\mathcal{I}$. Using (10), we see that $\psi \circ g = 0$ is equivalent to

$$ax - by + \alpha\bar{\tau} = 0, \quad a_0 = \alpha x_0 \quad \text{and} \quad b_0 = \alpha y_0.$$

Suppose that $\alpha = 1$. We have then

$$ay - bx = -\bar{\tau}, \quad a - 0 = x_0, \quad b_0 = y_0.$$

So we can write $a = x + \bar{A}$, $b = y + \bar{B}$, with $\bar{A}, \bar{B} \in \mathcal{I}$, and $\bar{\tau} = -y_0\bar{A} + x_0\bar{B}$.

Now we show that $\psi$ is injective. Let $(u, v, \lambda) \in 2\mathcal{O}_2 \oplus \mathfrak{m}\mathcal{I}$ be such that $\psi(u, v, \lambda) = 0$, i.e. $u(x + \bar{A}) + v(y + \bar{B}) + \lambda = 0$. Then we have $u_0x_0 + v_0y_0 = 0$, hence we can write $u = ey + \bar{\gamma}$, $v = -ex + \bar{\delta}$, with $e \in \mathcal{O}_2$, $\gamma, \delta \in \mathcal{I}$, and we obtain $-e_0\bar{\tau} + x\bar{\gamma} + y\bar{\delta} + \lambda = 0$, whence $(u, v, \lambda) = g(ey + \bar{\gamma}, -ex + \bar{\delta})$, and $\psi$ is injective.

Since $\psi$ is injective, $M$ is isomorphic to $\text{im}(\psi) \subset \mathcal{O}_2$, and $\text{im}(\psi) = (x + \bar{A}, y + \bar{B})$. This proves theorem 7.2.1. $\square$

### 7.3. PARAMETERIZATION OF $\mathcal{J}_P$

Let $x, y \in \mathcal{O}_2$ be such that $x_0, y_0$ are generators of $\mathfrak{m}$. If $\bar{A}, \bar{B} \in \mathcal{I}$, let $\tau_{x,y}(\bar{A}, \bar{B})$ be the image of $-y_0\bar{A} + x_0\bar{B}$ in $\mathfrak{m}\mathcal{I}/\mathfrak{m}^2\mathcal{I}$. By proposition 7.1.1,

$$\begin{gathered} \mathcal{J}_P \longrightarrow \mathfrak{m}\mathcal{I}/\mathfrak{m}^2\mathcal{I} \\ (x + \bar{A}, y + \bar{B}) \longmapsto \tau_{x,y}(\bar{A}, \bar{B}) \end{gathered}$$

is a bijection.

Let $x', y' \in \mathcal{O}_2$ be such that $x'_0, y'_0$ are generators of $\mathfrak{m}$. Then there exist $\bar{u}, \bar{v} \in \mathfrak{m}\mathcal{I}$ such that $(x', y') = (x + \bar{u}, y + \bar{v})$. There exists an invertible matrix $\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$ with coefficients in $\mathcal{O}_2$, such that

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \begin{pmatrix} x + \bar{u} \\ y + \bar{v} \end{pmatrix}.$$

Let $\Delta = \alpha\delta - \beta\gamma$. If $\bar{A}', \bar{B}' \in \mathcal{I}$, then $(x' + \bar{A}', y' + \bar{B}') = (x + \bar{A}, y + \bar{B})$ with

$$\bar{A} = \frac{\delta_0}{\Delta_0} \bar{A}' - \frac{\beta_0}{\Delta_0} \bar{B}' + \bar{u}, \quad \bar{B} = -\frac{\gamma_0}{\Delta_0} \bar{A}' + \frac{\alpha_0}{\Delta_0} \bar{B}' + \bar{v}.$$

It follows that

$$\tau_{x,y}(\bar{A}, \bar{B}) = \frac{1}{\Delta_0} \tau_{x',y'}(\bar{A}', \bar{B}') - y_0 \bar{u} + x_0 \bar{v},$$

We have $\mathfrak{m}\mathcal{I}/\mathfrak{m}^2\mathcal{I} \simeq \Omega_{X,P} \otimes L_P$. Let $\Delta(x, y)$ be the image of $x_0 \wedge y_0$ in $\omega_{X,P}$, and denote by $\Delta(x, y)^{-1}$ the corresponding element of $\omega_{X,P}^*$. For $\bar{A}, \bar{B} \in \mathcal{I}$ and $J = (x + \bar{A}, y + \bar{B})$, let

$$\lambda_{P,x,y}(J) = \Delta(x, y)^{-1} \otimes \tau_{x,y}(\bar{A}, \bar{B}) \in \omega_{X,P}^* \otimes \Omega_{X,P} \otimes L_P \simeq T_{X,P} \otimes L.$$
where $\iota$ is the inclusion. Hence we have

$$ (10) \qquad \mathrm{im}(g) = \{ (ey + \bar{\gamma}, -ex + \bar{\delta}, e_0\bar{\tau} + \rho(x_0\bar{\gamma} + y_0\bar{\delta})); e \in \mathcal{O}_2, \bar{\gamma}, \bar{\delta} \in \mathcal{J} \}. $$

In the exact sequence 8, $i: \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \to M$ comes from the inclusion

$$ \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \hookrightarrow 2\mathcal{O}_2 \oplus (\mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}), \text{ and } q: M \to \mathfrak{m} \text{ from } \phi_0. $$

It follows easily from (9) that $M$ is a $\mathcal{O}_1$-module (i.e. $\mathcal{J}M = \{0\}$) if and only if $\rho = 0$.

**Step 4. Properties of the extensions** – We will describe the morphism $h: \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J} \to \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$ induced by the canonical morphism $M \otimes_{\mathcal{O}_2} \mathcal{J} \to M$ and (8) ($M$ is balanced if and only if $h$ is an isomorphism). Let $\alpha \in \mathfrak{m}$, that we can write as $\alpha = \phi_0(a, b) = a_0x_0 + b_0y_0$, with $a, b \in \mathcal{O}_2$. Then from step 3, $h(\alpha \otimes \bar{t})$ is the unique $\beta \otimes \bar{t} \in \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$ such that $(0, 0, \beta \otimes \bar{t}) - (\bar{t}a, \bar{t}b, 0) \in g(K)$. Using (9), we see that $\beta = -\rho\alpha$, so that $h$ is the multiplication by $-\rho$.

Now suppose that $\rho \neq 0$. Then it is easy to see that $M^{(1)} = \mathrm{im}(i) = \mathfrak{m} \otimes_{\mathcal{O}_1} \mathcal{J}$, and that $M^{(1)}/M_1 \simeq \mathfrak{m}/\rho\mathfrak{m}$. It follows that $M$ is balanced if and only if $\rho$ is invertible, and that (8) is the canonical exact sequence $0 \to M^{(1)} \to M \to M^{(0)} \to 0$ if and only if $\rho = -1$.

**Step 5. Description of the morphisms** $M \to \mathcal{O}_2$ – We suppose that $\rho = -1$. Such a morphism comes from $\psi: 2\mathcal{O}_2 \oplus \mathfrak{m}\mathcal{J} \to \mathcal{O}_2$ such that $\psi \circ g = 0$. There exist $a, b \in \mathcal{O}_2, \alpha \in \mathcal{O}_1$, such that $\psi(u, v, \bar{w}) = au + bv + \alpha\bar{w}$ for every $u, v \in \mathcal{O}_2, \bar{w} \in \mathfrak{m}\mathcal{J}$. Using (10), we see that $\psi \circ g = 0$ is equivalent to

$$ ax - by + \alpha \bar{\tau} = 0, \quad a_0 = \alpha x_0 \quad \text{and} \quad b_0 = \alpha y_0. $$

Suppose that $\alpha = 1$. We have then

$$ ay - bx = -\bar{\tau}, \quad a - 0 = x_0, \quad b_0 = y_0. $$

So we can write $a = x + \bar{A}, b = y + \bar{B}$, with $\bar{A}, \bar{B} \in \mathcal{J}$, and $\bar{\tau} = -y_0\bar{A} + x_0\bar{B}$.

Now we show that $\psi$ is injective. Let $(u, v, \lambda) \in 2\mathcal{O}_2 \oplus \mathfrak{m}\mathcal{J}$ be such that $\psi(u, v, \lambda) = 0$, i.e. $u(x + \bar{A}) + v(y + \bar{B}) + \lambda = 0$. Then we have $u_0x_0 + v_0y_0 = 0$, hence we can write $u = ey + \bar{\gamma}, v = -ex + \bar{\delta}$, with $e \in \mathcal{O}_2, \gamma, \delta \in \mathcal{J}$, and we obtain $-e_0\bar{\tau} + x\bar{\gamma} + y\bar{\delta} + \lambda = 0$, whence $(u, v, \lambda) = g(ey + \bar{\gamma}, -ex + \bar{\delta})$, and $\psi$ is injective.

Since $\psi$ is injective, $M$ is isomorphic to $\mathrm{im}(\psi) \subset \mathcal{O}_2$, and $\mathrm{im}(\psi) = (x + \bar{A}, y + \bar{B})$. This proves theorem 7.2.1. $\square$

### 7.3. PARAMETERIZATION OF $\mathcal{J}_P$

Let $x, y \in \mathcal{O}_2$ be such that $x_0, y_0$ are generators of $\mathfrak{m}$. If $\bar{A}, \bar{B} \in \mathcal{J}$, let $\tau_{x,y}(\bar{A}, \bar{B})$ be the image of $-y_0\bar{A} + x_0\bar{B}$ in $\mathfrak{m}\mathcal{J}/\mathfrak{m}^2\mathcal{J}$. By proposition 7.1.2,

$$ \mathcal{J}_P \longrightarrow \mathfrak{m}\mathcal{J}/\mathfrak{m}^2\mathcal{J} \\ (x + \bar{A}, y + \bar{B}) \longmapsto \tau_{x,y}(\bar{A}, \bar{B}) $$

is a bijection.

Let $x', y' \in \mathcal{O}_2$ be such that $x'_0, y'_0$ are generators of $\mathfrak{m}$. Then there exist $\bar{u}, \bar{v} \in \mathfrak{m}\mathcal{J}$ such that $(x', y') = (x + \bar{u}, y + \bar{v})$. There exists an invertible matrix $\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$ with coefficients in $\mathcal{O}_2$, such
Then $\lambda_{P,x,y} : \mathcal{J}_P \to T_{X,P} \otimes L$ is a bijection. We have

$$ (11) \qquad \lambda_{P,x',y'}(J) = \lambda_{P,x,y}(J) + \Delta(x,y)^{-1} \otimes (-y_0 \bar{u} + x_0 \bar{v}) . $$

Hence for every $J_1, J_2 \in \mathcal{J}$ we have

$$ \lambda_{P,x,y}(J_1) - \lambda_{P,x,y}(J_2) = \lambda_{P,x',y'}(J_1) - \lambda_{P,x',y'}(J_2) . $$

It follows that

**7.3.1. Proposition:** $\mathcal{J}_P$ has a natural structure of affine space, with underlying $\mathbb{C}$-vector space $T_{X,P} \otimes L$.

## 7.4. IDEAL SHEAVES ON $X_2$

**7.4.1. Extension of ideals** – Let $J \in \mathcal{J}$. Then $J$ extends naturally to an ideal sheaf $J_2$ on $X_2$, $P$ being the unique point $Q \in X$ such that $J_{2,Q} \neq \mathcal{O}_{X_2,Q}$. More generally, let $Z$ be a smooth variety, $U$ an open subset of $X_2$, and $U_0 = U \cap X$. Let $\alpha : Z \to U_0$ be a morphism. Then $Z_\alpha = \{(z, \alpha(z)); z \in Z\}$ is a closed subvariety of $Z \times X$. Let $\mathcal{J}$ be a sheaf of ideals on $Z \times U$ such that the support of $\mathcal{O}_{Z \times U}/\mathcal{J}$ (i.e. the set of points where $\mathcal{J} \neq \mathcal{O}_{Z \times U}$) is $Z_\alpha$. The $\mathcal{J}$ extends naturally to a sheaf of ideals on $Z \times X_2$, such that the support of $\mathcal{O}_{Z \times X_2}/\mathcal{J}'$ is $Z_\alpha$.

**7.4.2. Families of ideals** – Suppose that

- The open subset $U_0$ is affine.

- There exists a section $\sigma : \mathcal{O}_X(U_0) \to \mathcal{O}_{X_2}(U)$ of the restriction $\mathcal{O}_{X_2}(U) \to \mathcal{O}_X(U_0)$, so we can view $\mathcal{O}_X(U_0)$ as a subring of $\mathcal{O}_{X_2}(U)$.

- there exist $x, y \in \mathcal{O}_X(U_0)$ such that for every $Q \in U_0$, $\mathfrak{m}_Q = (x - x(Q), y - y(Q))$ (where $\mathfrak{m}_Q$ is the maximal ideal of $\mathcal{O}_{X,Q}$), hence $\mathcal{O}_X(U_0)$ is generated by $dx$ and $dy$.

- $L|_U = \mathcal{J}_{X|U}$ is trivial. Let $\bar{t}$ be a generator of $L|_U$.

So we have a local description of $\mathcal{O}_{X|U_0}$:

- with differentials: $\mathcal{O}_X(U_0)$ is generated by $dx$ and $dy$.

- for every $Q \in U_0$, $\mathcal{O}_{X,Q} \simeq \mathfrak{m}_Q/\mathfrak{m}_Q^2$, and for every $\alpha, \beta \in \mathcal{O}_{X,Q}$, $\alpha dx + \beta dy$ corresponds to the image of $\alpha(Q)(x_0 - x_0(Q)) + \beta(Q)(y_0 - y_0(Q))$ in $\mathfrak{m}_Q/\mathfrak{m}_Q^2$.

Let $Y = U \times (U_0 \times \mathbb{C}^2)$, $p : Y \to U$, $p_0 : Y \to U_0$, $q_1, q_2 : Y \to \mathbb{C}$ be the projections. Then $(p^*(\sigma(x)) - p_0^*(x_0) + q_1\bar{t}, p^*(\sigma(y)) - p_0^*(y_0) + q_2\bar{t})$ is a regular sequence at each point of $Y$. Let

$$ \mathbf{I} = (p^*(\sigma(x)) - p_0^*(x_0) + q_1\bar{t}, p^*(\sigma(y)) - p_0^*(y_0) + q_2\bar{t}) , $$

which is an ideal sheaf on $Y$, of a subscheme $Z \subset Y$. One of the irreducible and connected components of $Z$, $Z_0$, contains the points $(P, P, u)$, $P \in U_0$, $u \in \mathbb{C}^2$. Let $\mathcal{J}^{\sigma, \bar{t}, x, y} = \mathcal{J}_{Z_0}$, which by 7.4.1 can be seen as a family of ideal sheaves on $X_2$ parameterized by $U_0 \times \mathbb{C}_2$ and flat on it. For every $(P, a, b) \in U_0 \times \mathbb{C}^2$, we have

$$ (\mathcal{J}^{\sigma, \bar{t}, x, y})_{P, a, b} = (\sigma(x) - x(P) + a\bar{t}, \sigma(y) - y(P) + b\bar{t}) . $$

We have an isomorphism

$$ \Theta^{x, y, \bar{t}} : \mathcal{O}_{U_0} \otimes \mathbb{C}^2 \to (\mathcal{O}_X \otimes \omega_X^* \otimes L)|_{U_0} = (T_X \otimes L)|_{U_0}, $$
that

$$ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \begin{pmatrix} x + \bar{u} \\ y + \bar{v} \end{pmatrix}. $$

Let $\Delta = \alpha\delta - \beta\gamma$. If $\overline{A'}, \overline{B'} \in \mathcal{J}$, then $(x' + \overline{A'}, y + \overline{B'}) = (x + \overline{A}, y + \overline{B})$ with

$$ \overline{A} = \frac{\delta_0}{\Delta_0} \overline{A}' - \frac{\beta_0}{\Delta_0} \overline{B}' + \bar{u}, \quad \overline{B} = -\frac{\gamma_0}{\Delta_0} \overline{A}' + \frac{\alpha_0}{\Delta_0} \overline{B}' + \bar{v}. $$

It follows that

$$ \tau_{x,y}(\overline{A}, \overline{B}) = \frac{1}{\Delta_0} \tau_{x',y'}(\overline{A'}, \overline{B}') - y_0 \bar{u} + x_0 \bar{v}, $$

We have $\mathfrak{m}\mathcal{J}/\mathfrak{m}^2\mathcal{J} \simeq \Omega_{X,P} \otimes L_P$. Let $\Delta(x, y)$ be the image of $x_0 \wedge y_0$ in $\omega_{X,P}$, and denote by $\Delta(x, y)^{-1}$ the corresponding element of $\omega_{X,P}^*$. For $\overline{A}, \overline{B} \in \mathcal{J}$ and $J = (x + \overline{A}, y + \overline{B})$, let

$$ \lambda_{P,x,y}(J) = \Delta(x, y)^{-1} \otimes \tau_{x,y}(\overline{A}, \overline{B}) \in \omega_{X,P}^* \otimes \Omega_{X,P} \otimes L_P \simeq T_{X,P} \otimes L. $$

Then $\lambda_{P,x,y} : \mathcal{J}_P \to T_{X,P} \otimes L$ is a bijection. We have

$$ (11) \qquad \lambda_{P,x',y'}(J) = \lambda_{P,x,y}(J) + \Delta(x,y)^{-1} \otimes (-y_0 \bar{u} + x_0 \bar{v}). $$

Hence for every $J_1, J_2 \in \mathcal{J}$ we have

$$ \lambda_{P,x,y}(J_1) - \lambda_{P,x,y}(J_2) = \lambda_{P,x',y'}(J_1) - \lambda_{P,x',y'}(J_2). $$

It follows that

**7.3.1. Proposition:** $\mathcal{J}_P$ has a natural structure of affine space, with underlying $\mathbb{C}$-vector space $T_{X,P} \otimes L$.

## 7.4. IDEAL SHEAVES ON $X_2$

Let $J \in \mathcal{J}_P$. Then $J$ extends naturally to an ideal sheaf $J_2$ on $X_2$, $P$ being the unique point $Q \in X$ such that $J_{2,Q} \neq \mathcal{O}_{X_2,Q}$. Hence we can see $\mathcal{J}_P$ as a set of ideal sheaves on $X_2$.

We will study $\mathbb{I} = \bigcup_{P \in X} \mathcal{J}_P$. We will see in 7.5 that $\mathbb{I}$ has a natural structure of an affine bundle on $X$, with associated vector bundle $T_X \otimes L$. To describe this structure we will see here how it is related to the construction of $X_2$ given in 4.

**7.4.1. Extension of ideals** – Let $Z$ be a smooth variety, $U$ an open subset of $X_2$, and $U_0 = U \cap X$. Let $\alpha : Z \to U_0$ be a morphism. Then $Z_\alpha = \{(z, \alpha(z)); z \in Z\}$ is a closed subvariety of $Z \times X$. Let $\mathcal{J}$ be a sheaf of ideals on $Z \times U$ such that the support of $\mathcal{O}_{Z \times U}/\mathcal{J}$ (i.e. the set of points where $\mathcal{J} \neq \mathcal{O}_{Z \times U}$) is $Z_\alpha$. Then $\mathcal{J}$ extends naturally to a sheaf of ideals $\mathcal{J}'$ on $Z \times X_2$, such that the support of $\mathcal{O}_{Z \times X_2}/\mathcal{J}'$ is $Z_\alpha$.

## 7.4.2. Families of ideals – Suppose that

- The open subset $U_0$ is affine.

- There exists a section $\sigma : \mathcal{O}_X(U_0) \to \mathcal{O}_{X_2}(U)$ of the restriction $\mathcal{O}_{X_2}(U) \to \mathcal{O}_X(U_0)$, so we can view $\mathcal{O}_X(U_0)$ as a subring of $\mathcal{O}_{X_2}(U)$.
defined by

$$ \Theta^{x,y,\bar{t}}(a,b) = (-a.dy + b(dx)) \otimes (dx \wedge dy)^{-1} \otimes \bar{t}, $$

for every $a, b \in \mathcal{O}_X(U_0)$. Let

$$ \mathcal{J}^{\sigma, \bar{t}, x, y} = ((\Theta^{x, y, \bar{t}})^{-1})^{\sharp}(\mathcal{J}^{\sigma, \bar{t}, x, y}), $$

so that $(\mathcal{J}^{\sigma, \bar{t}, x, y})_{P, \Theta^{x, y, \bar{t}}(a,b)} = (\mathcal{J}^{\sigma, \bar{t}, x, y})_{P, a,b}$.

**7.4.3. Proposition: $\mathcal{J}^{\sigma, \bar{t}, x, y}$ is independent of $x, y$ and $\bar{t}$**.

*Proof.* For every $Q \in U_0$ we have

$$ (\Theta^{x,y,\bar{t}}(a,b))_Q = \lambda_{Q,\sigma(x),\sigma(y)}((\mathcal{J}^{\sigma,\bar{t},x,y})_{Q,\Theta^{x,y,\bar{t}}(a,b)}). $$

In other words, for every $\eta \in (\Omega_X \otimes \omega_X^* \otimes L)_Q$,

$$ \lambda_{Q,\sigma(x),\sigma(y)}((\mathcal{J}^{\sigma,\bar{t},x,y})_{Q,\eta}) = \eta. $$

If $x, y$ are replaced with $x', y' \in \mathcal{O}_X(U_0)$, we have, for every $P \in U_0$

$$ (\sigma(x) - x(P), \sigma(y) - y(P)) = (\sigma(x') - x'(P), \sigma(y') - y'(P)) $$

(it is the ideal of $\mathcal{O}_{X_2,P}$ generated by $\sigma(\mathfrak{m}_Q)$). Hence, by formula (11), the maps $\lambda_{Q,\sigma(x),\sigma(y)}$ and $\lambda_{Q,\sigma(x'),\sigma(y')}$ are the same, and since they are isomorphisms, $\mathcal{J}^{\sigma, \bar{t}, x, y}$ is independent of the choice of $x, y$ and $\bar{t}$. $\square$

We will note $\mathcal{J}^{\sigma} = \mathcal{J}^{\sigma, \bar{t}, x, y}$. It is a flat family of ideal sheaves on $X_2$, parameterized by $(T_X \otimes L)|_{U_0}$.

The map

$$ f_Q : T_{X,Q} \otimes L_Q = (\Omega_X \otimes \omega_X^* \otimes L)_Q \longrightarrow \mathcal{J}_Q \\ \eta \longleftarrow \mathcal{J}^{\sigma})_{Q,\eta} $$

is a bijection, i.e. $(\Omega_X \otimes \omega_X^* \otimes L)_Q$ parameterizes the set of ideal sheaves of $\mathcal{O}_{X_2}$, which are balanced and whose restriction to $X$ is the ideal sheaf of $Q$.

Let $\mathcal{J}_{U_0} = \bigcup_{Q \in U_0} \mathcal{J}_Q$. Then $\mathcal{J}_{U_0}$ parameterizes the set of ideal sheaves of $\mathcal{O}_{X_2}$, which are balanced and whose restriction to $X$ is the ideal sheaf of a point of $U_0$. We have a bijection

$$ \mu_{\sigma} : (T_X \otimes L)|_{U_0} \longrightarrow \mathcal{J}_{U_0} $$

over $U_0$ defined by the maps $f_Q$.

**7.4.4. – Change of parameters –** Suppose that we have another section $\sigma' : \mathcal{O}_X(U_0) \to \mathcal{O}_{X_2}(U)$ of the restriction $\mathcal{O}_{X_2}(U) \to \mathcal{O}_X(U_0)$. Then there is a derivation $D$ of $\mathcal{O}_X(U_0)$ such that

$$ \sigma'(\lambda) = \sigma(\lambda) + D(\lambda)\bar{t}, $$

for every $\lambda \in \mathcal{O}_X(U_0)$. Let

$$ x' = \sigma'(x) = \sigma(x) + D(x)\bar{t}, \quad y' = \sigma'(y) = \sigma(y) + D(y)\bar{t}. $$

We can view $D \otimes \bar{t}$ as a section of $(T_X \otimes L)|_{U_0}$.
- there exist $x, y \in \mathcal{O}_X(U_0)$ such that for every $Q \in U_0$, $\mathfrak{m}_Q = (x - x(Q), y - y(Q))$ (where $\mathfrak{m}_Q$ is the maximal ideal of $\mathcal{O}_{X,Q}$), hence $\mathcal{O}_X(U_0)$ is generated by $dx$ and $dy$.

- $L_{|U} = \mathcal{I}_{X|U}$ is trivial. Let $\bar{t}$ be a generator of $L_{|U}$.

So we have a local description of $\mathcal{O}_{X|U_0}$:

- with differentials: $\mathcal{O}_X(U_0)$ is generated by $dx$ and $dy$.

- for every $Q \in U_0$, $\mathcal{O}_{X,Q} \simeq \mathfrak{m}_Q/\mathfrak{m}_Q^2$, and for every $\alpha, \beta \in \mathcal{O}_{X,Q}$, $\alpha dx + \beta dy$ corresponds to the image of $\alpha(Q)(x_0 - x_0(Q)) + \beta(Q)(y_0 - y_0(Q))$ in $\mathfrak{m}_Q/\mathfrak{m}_Q^2$.

Let $Y = U \times (U_0 \times \mathbb{C}^2)$, $p:Y \to U$, $p_0:Y \to U_0$, $q_1, q_2:Y \to \mathbb{C}$ be the projections. Then $(p^*(\sigma(x)) - p_0^*(x_0) + q_1\bar{t}, p^*(\sigma(y)) - p_0^*(y_0) + q_2\bar{t})$ is a regular sequence at each point of $Y$. Let

$$ \mathbf{I} = (p^*(\sigma(x)) - p_0^*(x_0) + q_1\bar{t}, p^*(\sigma(y)) - p_0^*(y_0) + q_2\bar{t}), $$

which is an ideal sheaf on $Y$, of a subscheme $Z \subset Y$. One of the irreducible and connected components of $Z$, $Z_0$, contains the points $(P, P, u)$, $P \in U_0$, $u \in \mathbb{C}^2$. Let $\mathcal{J}^{\sigma, \bar{t}, x, y} = \mathcal{J}_{Z_0}$, which by 7.4.1 can be seen as a family of ideal sheaves on $X_2$ parameterized by $U_0 \times \mathbb{C}_2$ and flat on it. For every $(P, a, b) \in U_0 \times \mathbb{C}^2$, we have

$$ (\mathcal{J}^{\sigma, \bar{t}, x, y})_{P, a, b} = (\sigma(x) - x(P) + a\bar{t}, \sigma(y) - y(P) + b\bar{t}). $$

We have an isomorphism

$$ \Theta^{x,y,\bar{t}} : \mathcal{O}_{U_0} \otimes \mathbb{C}^2 \to (\mathcal{O}_X \otimes \omega_X^* \otimes L)|_{U_0} = (T_X \otimes L)|_{U_0}, $$

defined by

$$ \Theta^{x,y,\bar{t}}(a,b) = (-a.dy + b(dx)) \otimes (dx \wedge dy)^{-1} \otimes \bar{t}, $$

for every $a, b \in \mathcal{O}_X(U_0)$. Let

$$ \mathcal{J}^{\sigma, \bar{t}, x, y} = ((\Theta^{x,y,\bar{t}})^{-1})^{\sharp}(\mathcal{J}^{\sigma, \bar{t}, x, y}), $$

so that $(\mathcal{J}^{\sigma, \bar{t}, x, y})_{P, \Theta^{x,y,\bar{t}}(a,b)} = (\mathcal{J}^{\sigma, \bar{t}, x, y})_{P, a, b}$.

**7.4.3. Proposition: $\mathcal{J}^{\sigma, \bar{t}, x, y}$ is independent of $x, y$ and $\bar{t}$**

*Proof.* For every $Q \in U_0$ we have

$$ (\Theta^{x,y,\bar{t}}(a,b))_Q = \lambda_{Q,\sigma(x),\sigma(y)}((\mathcal{J}^{\sigma,\bar{t},x,y})_{Q,\Theta^{x,y,\bar{t}}(a,b)}). $$

In other words, for every $\eta \in (\mathcal{O}_X \otimes \omega_X^* \otimes L)_Q$,

$$ \lambda_{Q,\sigma(x),\sigma(y)}((\mathcal{J}^{\sigma,\bar{t},x,y})_{Q,\eta}) = \eta. $$

If $x, y$ are replaced with $x', y' \in \mathcal{O}_X(U_0)$, we have, for every $P \in U_0$

$$ (\sigma(x) - x(P), \sigma(y) - y(P)) = (\sigma(x') - x'(P), \sigma(y') - y'(P)) $$

(it is the ideal of $\mathcal{O}_{X_2,P}$ generated by $\sigma(\mathfrak{m}_Q)$). Hence, by formula (11), the maps $\lambda_{Q,\sigma(x),\sigma(y)}$ and $\lambda_{Q,\sigma(x'),\sigma(y')}$ are the same, and since they are isomorphisms, $\mathcal{J}^{\sigma, \bar{t}, x, y}$ is independent of the choice of $x, y$ and $\bar{t}$. $\square$
**7.4.5. Proposition:** For every $Q \in U_0$, $u \in T_{X,Q} \otimes L_Q$,

$$ \mu_{\sigma'}(u) = \mu_{\sigma}(u + (D \otimes \bar{t})(Q)). $$

*Proof.* From the definitions, using $x', y'$ instead of $x, y$, we have

$$ \begin{aligned} (\mathcal{J}^{\sigma'})_{P, \Theta^{x', y', \bar{t}}(a, b)} &= (\sigma(x) - x(P) + (a + D(P)(x))\bar{t}, \sigma(y) - y(P) + (b + D(P)(y))\bar{t}), \\ &= (\mathcal{J}^{\sigma})_{P, \Theta^{x, y, \bar{t}}(a + D(P)(x), b + D(P)(y))}, \end{aligned} $$

which is equivalent to

$$ \begin{aligned} \mu_{\sigma', \bar{t}}((-a dy + b dx) \otimes (dx \wedge dy)^{-1} \otimes \bar{t}) &= \mu_{\sigma, \bar{t}}(-(a + D(x))dy + (b + D(y))dx) \otimes (dx \wedge dy)^{-1} \otimes \bar{t}. \end{aligned} $$

Using the canonical isomorphism $\omega_{X,P}^* \otimes \Omega_{X,P} \simeq T_{X,P}$, one obtains proposition 7.4.5. $\square$

**7.5. MODULI SPACES OF IDEALS AND UNIVERSAL SHEAVES**

**7.5.1. Ideal sheaves on X** – Let $\Delta \subset X \times X$ be the diagonal. Then $\mathcal{J}_{\Delta}$ can be seen as a family of ideal sheaves on $X$, parameterized by $X$. For every $x \in X$

$$ \mathcal{J}_{\Delta, x} = \mathcal{J}_{\Delta|_{X \times \{x\}}} = \mathcal{J}_x. $$

**7.5.2. Construction of the moduli space and the universal sheaf** – Let $(V_i)_{1 \le i \le k}$ be an open affine cover of $X$ such that for each $V_i$ the conditions of 7.4.2 are satisfied: we have a section $\sigma_i : \mathcal{O}_X(V_i) \to \mathcal{O}_{X_2}(V_i)$ of the restriction $\mathcal{O}_{X_2}(V_i) \to \mathcal{O}_X(V_i)$, and $\bar{t}_i$ is a generator of $L|_{V_i}$. We obtain the family $\mathcal{J}^{\sigma_i}$ of ideal sheaves on $X_2$ parameterized by $(T_X \otimes L)|_{V_i}$. If $1 \le i < j \le k$ we have

$$ \sigma_{j|V_{ij}} = \sigma_{i|V_{ij}} + D_{ij} \bar{t}_i, $$

where $D_{ij}$ is a derivation of $\mathcal{O}_X(V_{ij})$. With the notations of 4, $(\alpha_i^{-1} \otimes D_{ij})_{1 \le i < j \le k}$ represents an element $\lambda$ of $H^1(X, T_X \otimes L)$. From 4.2, $\mathbb{P}(H^1(X, T_X \otimes L)) \cup \{0\}$ parameterizes the set of primitive double schemes $Y$ such that $Y_{red} = X$ and associated line bundle $L$, and $\mathbb{C}\lambda = \zeta(X_2)$ is the element corresponding to $X_2$.

We have an affine isomorphism

$$ \rho_{ij} : (T_X \otimes L)|_{V_{ij}} \longrightarrow (T_X \otimes L)|_{V_{ij}} \\ u \longleftarrow \longrightarrow u + D_{ij}(u) \otimes \bar{t}_i $$

and

$$ \rho_{ij}^{\sharp}(\mathcal{J}^{\sigma_i}) \simeq \mathcal{J}^{\sigma_j}. $$

The $\rho_{ij}$ define an affine bundle $\mathbb{I}$ on $X$, with associated vector bundle $T_X \otimes L$. According to 2.4 we have

**7.5.3. Proposition:** The element $\eta(\mathbb{I})$ of $(\mathbb{P}(H^1(X, T_X \otimes L)) \cup \{0\}) / \text{Aut}(T_X \otimes L)$ associated to $\mathbb{I}$ is the image of $\zeta(X_2)$.
We will note $\mathcal{J}^{\sigma} = \mathcal{J}^{\sigma, \bar{t}, x, y}$. It is a flat family of ideal sheaves on $X_2$, parameterized by $(T_X \otimes L)|_{U_0}$.

The map

$$f_Q : T_{X,Q} \otimes L_Q = (\Omega_X \otimes \omega_X^* \otimes L)_Q \longrightarrow \mathcal{J}_Q$$
$$\eta \longmapsto (\mathcal{J}^\sigma)_{Q,\eta}$$

is a bijection, i.e. $(\Omega_X \otimes \omega_X^* \otimes L)_Q$ parameterizes the set of ideal sheaves of $\mathcal{O}_{X_2}$, which are balanced and whose restriction to $X$ is the ideal sheaf of $Q$.

Let $\mathcal{J}_{U_0} = \bigcup_{Q \in U_0} \mathcal{J}_Q$. Then $\mathcal{J}_{U_0}$ parameterizes the set of ideal sheaves of $\mathcal{O}_{X_2}$, which are balanced and whose restriction to $X$ is the ideal sheaf of a point of $U_0$. We have a bijection

$$\mu_{\sigma} : (T_X \otimes L)|_{U_0} \longrightarrow \mathcal{J}_{U_0}$$

over $U_0$ defined by the maps $f_Q$.

**7.4.4. - Change of parameters** – Suppose that we have another section $\sigma' : \mathcal{O}_X(U_0) \to \mathcal{O}_{X_2}(U)$ of the restriction $\mathcal{O}_{X_2}(U) \to \mathcal{O}_X(U_0)$. Then there is a derivation $D$ of $\mathcal{O}_X(U_0)$ such that

$$\sigma'(\lambda) = \sigma(\lambda) + D(\lambda)\bar{t},$$

for every $\lambda \in \mathcal{O}_X(U_0)$. Let

$$x' = \sigma'(x) = \sigma(x) + D(x)\bar{t}, \quad y' = \sigma'(y) = \sigma(y) + D(y)\bar{t}.$$

We can view $D \otimes \bar{t}$ as a section of $(T_X \otimes L)|_{U_0}$.

**7.4.5. Proposition:** For every $Q \in U_0$, $u \in T_{X,Q} \otimes L_Q$,

$$\mu_{\sigma'}(u) = \mu_{\sigma}(u + (D \otimes \bar{t})(Q)).$$

*Proof.* From the definitions, using $x'$, $y'$ instead of $x$, $y$, we have

$$\begin{aligned}
(\mathcal{J}^{\sigma'})_{P, \Theta^{x', y', \bar{t}}(a,b)} &= (\sigma(x) - x(P) + (a + D(P)(x))\bar{t}, \sigma(y) - y(P) + (b + D(P)(y))\bar{t}), \\
&= (\mathcal{J}^{\sigma})_{P, \Theta^{x, y, \bar{t}}(a+D(P)(x), b+D(P)(y))},
\end{aligned}$$

which is equivalent to

$$\mu_{\sigma', \bar{t}}((-a dy + b dx) \otimes (dx \wedge dy)^{-1} \otimes \bar{t}) = \\ \mu_{\sigma, \bar{t}}(-(a + D(x))dy + (b + D(y))dx) \otimes (dx \wedge dy)^{-1} \otimes \bar{t}.$$

Using the canonical isomorphism $\omega_{X,P}^* \otimes \mathcal{O}_{X,P} \simeq T_{X,P}$, one obtains proposition 7.4.5. $\square$

<figure><img src="image_5.png" /></figure>
The closed points of $\mathbb{I}$ are the ideal sheaves $\mathcal{I}$ on $X_2$ of subschemes $Z$ such that

- $Z$ contains only one closed point $P$ of $X$.

- there exist $x, y \in \mathcal{O}_{X_2,P}$ such that their images in $\mathcal{O}_{X,P}$ generate the maximal ideal, and $\mathcal{I}_P = (x, y)$.

The sheaves $\mathcal{I}^{\sigma_i}$ can be glued, using 7.4.4, to obtain a flat family of ideal sheaves $\mathcal{I}$ parameterized by $\mathbb{I}$. It is a balanced sheaf on $X_2 \times \mathbb{I}$. If $p_X : \mathbb{I} \to X$ is the projection, we have

$$ \mathcal{I}_{|X \times \mathbb{I}} \simeq p_X^{\sharp}(\mathcal{I}_{\Delta}). $$

## 7.6. MODULI SPACES OF SHEAVES

The ideal sheaves $\mathcal{I} \in \mathbb{I}$ can also be deformed by tensoring with a line bundle. This is why, to obtain a moduli space of sheaves, one must consider the sheaves $\mathcal{I} \otimes D, \mathcal{I} \in \mathbb{I}, D \in \mathrm{Pic}(X_2)$.

**7.6.1. Moduli spaces on rank 1 sheaves on $X$** – Let $\mathbb{D}$ be a Poincaré bundle on $\mathrm{Pic}(X)$ and $\mathbb{M}_X = X \times \mathrm{Pic}(X)$. Let $p_1 : \mathbb{M}_X \to X, p_2 : \mathbb{M}_X \to \mathrm{Pic}(X)$ be the projections, and

$$ \mathfrak{D} = p_1^{\sharp}(\mathcal{I}_{\Delta}) \otimes p_2^{\sharp}(\mathbb{D}). $$

It is a flat family of sheaves on $X$ parameterized by $\mathbb{M}_X$. For every $(x, D) \in \mathbb{M}_X$, $\mathfrak{D}_{(x,D)} = \mathcal{I}_x \otimes D$.

The following properties are well known:

- Let $(x, D), (x', D') \in \mathbb{M}_X$. Suppose that $\mathfrak{D}_{(x,D)} \simeq \mathfrak{D}_{(x',D')}$. Then $(x, D) = (x', D')$. So we can see $\mathbb{M}_X$ as the family of sheaves $\mathfrak{D}_{(x,D)}$.

- Let $\mathcal{F}$ be a flat family of sheaves on $X$ parameterized by a scheme $Y$. Let $y \in Y$ be a closed point such that $\mathcal{F}_y \in \mathbb{M}_X$. Then there exists an open neighborhood $V$ of $y$ such that for every closed point $v \in V$, $\mathcal{F}_v \in \mathbb{M}_X$.

- Let $\mathcal{F}$ be a flat family of sheaves on $X$ parameterized by a scheme $Y$. Suppose that for every closed point $y \in Y$, $\mathcal{F}_y \in \mathbb{M}_X$. Then there exists a unique morphism $f_{\mathcal{F}} : Y \to \mathbb{M}_X$, and a line bundle $\Gamma$ on $Y$ such that $\mathcal{F} \simeq f_{\mathcal{F}}^{\sharp}(\mathfrak{D}) \otimes p_Y^{*}(\Gamma)$ (where $p_Y = X \times Y \to Y$ is the projection).

Thus $\mathbb{M}_X$ is a moduli space of rank 1 sheaves on $X$ (cf. 2.6).

**7.6.2. Moduli spaces on rank 1 sheaves on $X_2$** – We can do the same work on $X_2$, using $\mathbb{I}$ and $\mathrm{Pic}(X_2)$.

**7.6.3. Lemma:** Let $\mathcal{I}, \mathcal{I}'$ be ideals in $\mathbb{I}$, and $\mathbb{D}$ a line bundle on $X_2$. If $\mathcal{I}' \simeq \mathcal{I} \otimes \mathbb{D}$ then $\mathbb{D} = \mathcal{O}_{X_2}$ and $\mathcal{I}' = \mathcal{I}$.

*Proof.* This follows easily from the fact that $\mathcal{I}^* = \mathcal{I}'^* = \mathcal{O}_{X_2}$. $\square$

Let $\mathbf{I}(X_2) = \mathbb{I} \times \mathrm{Pic}(X_2)$. Let $q_1 : \mathbf{I}(X_2) \to \mathbb{I}, q_2 : \mathbf{I}(X_2) \to \mathrm{Pic}(X_2)$ be the projections. Let $(P_i)_{i \in I}$ be an open cover of $\mathrm{Pic}(X_2)$ such that for every $i \in I$, there is a Poincaré bundle $\mathcal{D}_i$ on $X_2 \times P_i$, which define $\mathrm{Pic}(X_2)$ as a *fine moduli space* of sheaves (cf. 2.6, 4.6).
## 7.5. MODULI SPACES OF IDEALS AND UNIVERSAL SHEAVES

Using the results of 7.4 we show that the set of ideals $\mathbb{I}$ has a natural structure of an affine bundle on $X$, with associated vector bundle $T_X \otimes L$. We construct a *universal sheaf* on $\mathbb{I} \times X_2$.

**7.5.1.** *Ideal sheaves on X* – Let $\Delta \subset X \times X$ be the diagonal. Then $\mathcal{J}_\Delta$ can be seen as a family of ideal sheaves on $X$, parameterized by $X$. For every $x \in X$

$$ \mathcal{J}_{\Delta, x} = \mathcal{J}_{\Delta|_{X \times \{x\}}} = \mathcal{J}_x. $$

**7.5.2.** *Construction of the moduli space and the universal sheaf* – Let $(V_i)_{1\le i\le k}$ be an open affine cover of $X$ such that for each $V_i$ the conditions of 7.4.2 are satisfied: we have a section $\sigma_i : \mathcal{O}_X(V_i) \to \mathcal{O}_{X_2}(V_i)$ of the restriction $\mathcal{O}_{X_2}(V_i) \to \mathcal{O}_X(V_i)$, and $\bar{t}_i$ is a generator of $L|_{V_i}$. We obtain the family $\mathcal{J}^{\sigma_i}$ of ideal sheaves on $X_2$ parameterized by $(T_X \otimes L)|_{V_i}$. If $1 \le i < j \le k$ we have

$$ \sigma_{j|V_{ij}} = \sigma_{i|V_{ij}} + D_{ij} \cdot \bar{t}_i, $$

where $D_{ij}$ is a derivation of $\mathcal{O}_X(V_{ij})$. With the notations of 4, $(\alpha_i^{-1} \otimes D_{ij})_{1\le i<j\le k}$ represents an element $\lambda$ of $H^1(X, T_X \otimes L)$. From 4.2, $\mathbb{P}(H^1(X, T_X \otimes L)) \cup \{0\}$ parameterizes the set of primitive double schemes $Y$ such that $Y_{red} = X$ and associated line bundle $L$, and $\mathbb{C}\lambda = \zeta(X_2)$ is the element corresponding to $X_2$.

We have an affine isomorphism

$$ \rho_{ij} : (T_X \otimes L)|_{V_{ij}} \longrightarrow (T_X \otimes L)|_{V_{ij}} \\ u \longleftarrow u + D_{ij}(u) \otimes \bar{t}_i $$

and

$$ \rho_{ij}^{\sharp}(\mathcal{J}^{\sigma_i}) \simeq \mathcal{J}^{\sigma_j}. $$

The $\rho_{ij}$ define an affine bundle $\mathbb{I}$ on $X$, with associated vector bundle $T_X \otimes L$. According to 2.4 we have

**7.5.3. Proposition: The element $\eta(\mathbb{I})$ of $(\mathbb{P}(H^1(X, T_X \otimes L)) \cup \{0\}) / \text{Aut}(T_X \otimes L)$ associated to $\mathbb{I}$ is the image of $\zeta(X_2)$**.

The closed points of $\mathbb{I}$ are the ideal sheaves $\mathcal{J}$ on $X_2$ of subschemes $Z$ such that

- $Z$ contains only one closed point $P$ of $X$.

- there exist $x, y \in \mathcal{O}_{X_2, P}$ such that their images in $\mathcal{O}_{X, P}$ generate the maximal ideal, and $\mathcal{J}_P = (x, y)$.

The sheaves $\mathcal{J}^{\sigma_i}$ can be glued, using 7.4.4, to obtain a flat family of ideal sheaves $\mathcal{J}$ parameterized by $\mathbb{I}$. It is a balanced sheaf on $X_2 \times \mathbb{I}$. If $p_X : \mathbb{I} \to X$ is the projection, we have

$$ \mathcal{J}_{|X \times \mathbb{I}} \simeq p_X^{\sharp}(\mathcal{J}_{\Delta}). $$
Let $\mathcal{H}_i = q_1^\sharp(\mathcal{J}) \otimes q_2^\sharp(\mathcal{D}_i)$, which is a sheaf on $\mathbb{I} \times P_i$. For every ideal $\mathcal{J}$ in $\mathbb{I}$ and $D \in P_i$, $\mathcal{H}_{\mathcal{J},D} = \mathcal{J} \otimes \mathcal{D}_{i,D}$.

**7.6.4. Proposition:** Let $Y$ be a scheme and $\mathcal{E}$ a coherent sheaf on $X_2 \times Y$, flat over $Y$. Let $y \in Y$ be a closed point such that $\mathcal{E}_y \in \mathcal{I}(X_2)$. Then there exists an neighborhood $V \subset Y$ of $y$ such that

- $\mathcal{E}_{1|_{X \times V}}$ and $\mathcal{E}_{|_{X \times V}}$ are flat on $V$.

- For every closed point $v \in V$, $\mathcal{E}_v \in \mathcal{I}(X_2)$, and $(\mathcal{E}_1)_v = (\mathcal{E}_v)_1$, $(\mathcal{E}_{|_{X \times V}})_v = (\mathcal{E}_v)|_X$.

*Proof.* This follows from theorem 6.2.1, 1-.

□

The proof of the following result is similar of that of theorem 1.1.3 of [15], and shows that $\mathcal{I}(X_2)$ is a moduli space of sheaves in the sense of 2.6.

**7.6.5. Theorem:** Let $Y$ be a scheme and $\mathcal{E}$ a coherent sheaf on $X_2 \times Y$, flat over $Y$, such that for every closed point $y \in Y$, $\mathcal{E}_y \in \mathcal{I}(X_2)$. Then there exists a unique morphism $f_{\mathcal{E}} : Y \to \mathcal{I}(X_2)$ and an open cover $(Y_i)_{i \in I}$ such that for every $i \in I$, $f_{\mathcal{E}}^\sharp(\mathcal{H})|_{X_2 \times Y_i} \simeq \mathcal{E}|_{X_2 \times Y_i}$.

Let $\Gamma(X_2) \subset \mathrm{Pic}(X)$ be the algebraic subgroup of vector bundles that can be extended to $X_2$. According to 4.6, $\mathrm{Pic}(X_2)$ is an affine bundle on $\Gamma(X_2)$ with associated vector bundle $\mathcal{O}_{\Gamma(X_2)} \otimes H^1(X, L)$. Let $p_X : X \times \Gamma(X_2) \to X$, be the projection. Then $\mathcal{I}(X_2)$ is an affine bundle on $X \times \Gamma(X_2)$, with associated vector bundle $(\mathcal{O}_{X \times \Gamma(X_2)} \otimes H^1(X, L)) \oplus p_X^*(T_X \otimes L)$.

## REFERENCES

[1] Bangere, P., Mukherjee, J., Raychaudhry, D. *K3 carpets on minimal rational surfaces and their smoothing*. Internat. J. Math. 32 (2021), no. 6, Paper No. 2150032

[2] Bănică, C., Forster, O. *Multiple structures on space curves*. In: Sundararaman, D. (Ed.) Proc. of Lefschetz Centennial Conf. (10-14 Dec. Mexico), Contemporary Mathematics 58, AMS, 1986, 47-64.

[3] Bayer, D., Eisenbud, D. *Ribbons and their canonical embeddings*. Trans. of the Amer. Math. Soc., 1995, 347-3, 719-756.

[4] Chen, D., Kass, J. L. *Moduli of generalized line bundles on a ribbon*. J. of Pure and Applied Algebra 220 (2016), 822-844.

[5] Drézet, J.-M. *Faisceaux cohérents sur les courbes multiples*. Collect. Math. 2006, 57-2, 121-171.

[6] Drézet, J.-M. *Paramétrisation des courbes multiples primitives*. Adv. in Geom. 2007, 7, 559-612.

[7] Drézet, J.-M. *Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives*. Mathematische Nachrichten, 2009, 282-7, 919-952.

[8] Drézet, J.-M. *Sur les conditions d'existence des faisceaux semi-stables sur les courbes multiples primitives*. Pacific Journ. of Math. 2011, 249-2, 291-319.

[9] Drézet, J.-M. *Courbes multiples primitives et déformations de courbes lisses*. Annales de la Faculté des Sciences de Toulouse 22, 1 (2013), 133-154.

[10] Drézet, J.-M. *Fragmented deformations of primitive multiple curves*. Central European Journal of Mathematics 11, no 12 (2013), 2106-2137.

[11] Drézet, J.-M. *Reducible deformations and smoothing of primitive multiple curves*. Manuscripta Mathematica 148 (2015), 447-469.
## 7.6. MODULI SPACES OF SHEAVES

The ideal sheaves $\mathcal{I} \in \mathbb{I}$ can also be deformed by tensoring with a line bundle. This is why one must consider also the sheaves $\mathcal{I} \otimes D$, $\mathcal{I} \in \mathbb{I}$, $D \in \mathrm{Pic}(X_2)$. In this way we obtain here a moduli space of sheaves (with a suitable universal property described in proposition 7.6.4), isomorphic to $\mathbb{I} \times \mathrm{Pic}(X_2)$.

**7.6.1. Moduli spaces on rank 1 sheaves on X** – Let $\mathbb{D}$ be a Poincaré bundle on $\mathrm{Pic}(X)$ and $\mathbb{M}_X = X \times \mathrm{Pic}(X)$. Let $p_1 : \mathbb{M}_X \to X$, $p_2 : \mathbb{M}_X \to \mathrm{Pic}(X)$ be the projections, and

$$ \mathfrak{D} = p_1^{\sharp}(\mathcal{I}_{\Delta}) \otimes p_2^{\sharp}(\mathbb{D}) . $$

It is a flat family of sheaves on $X$ parameterized by $\mathbb{M}_X$. For every $(x, D) \in \mathbb{M}_X$, $\mathfrak{D}_{(x,D)} = \mathcal{I}_x \otimes D$.

The following properties are well known:

- Let $(x, D), (x', D') \in \mathbb{M}_X$. Suppose that $\mathfrak{D}_{(x,D)} \simeq \mathfrak{D}_{(x',D')}$. Then $(x, D) = (x', D')$. So we can see $\mathbb{M}_X$ as the family of sheaves $\mathfrak{D}_{(x,D)}$.

- Let $\mathcal{F}$ be a flat family of sheaves on $X$ parameterized by a scheme $Y$. Let $y \in Y$ be a closed point such that $\mathcal{F}_y \in \mathbb{M}_X$. Then there exists an open neighborhood $V$ of $y$ such that for every closed point $v \in V$, $\mathcal{F}_v \in \mathbb{M}_X$.

- Let $\mathcal{F}$ be a flat family of sheaves on $X$ parameterized by a scheme $Y$. Suppose that for every closed point $y \in Y$, $\mathcal{F}_y \in \mathbb{M}_X$. Then there exists a unique morphism $f_{\mathcal{F}} : Y \to \mathbb{M}_X$, and a line bundle $\Gamma$ on $Y$ such that $\mathcal{F} \simeq f_{\mathcal{F}}^{\sharp}(\mathfrak{D}) \otimes p_Y^{\ast}(\Gamma)$ (where $p_Y = X \times Y \to Y$ is the projection).

Thus $\mathbb{M}_X$ is a moduli space of rank 1 sheaves on $X$ (cf. 2.6).

**7.6.2. Moduli spaces on rank 1 sheaves on $X_2$** – We can do the same work on $X_2$, using $\mathbb{I}$ and $\mathrm{Pic}(X_2)$.

**7.6.3. Lemma:** Let $\mathcal{I}$, $\mathcal{I}'$ be ideals in $\mathbb{I}$, and $\mathbb{D}$ a line bundle on $X_2$. If $\mathcal{I}' \simeq \mathcal{I} \otimes \mathbb{D}$ then $\mathbb{D} = \mathcal{O}_{X_2}$ and $\mathcal{I}' = \mathcal{I}$.

*Proof.* This follows easily from the fact that $\mathcal{I}^* = \mathcal{I}'^* = \mathcal{O}_{X_2}$. $\square$

Let $\mathcal{I}(X_2) = \mathbb{I} \times \mathrm{Pic}(X_2)$. Let $q_1 : \mathcal{I}(X_2) \to \mathbb{I}$, $q_2 : \mathcal{I}(X_2) \to \mathrm{Pic}(X_2)$ be the projections. Let $(P_i)_{i \in I}$ be an open cover of $\mathrm{Pic}(X_2)$ such that for every $i \in I$, there is a Poincaré bundle $\mathcal{D}_i$ on $X_2 \times P_i$, which define $\mathrm{Pic}(X_2)$ as a *fine moduli space* of sheaves (cf. 2.6, 4.6).

Let $\mathcal{H}_i = q_1^{\sharp}(\mathcal{I}) \otimes q_2^{\sharp}(\mathcal{D}_i)$, which is a sheaf on $\mathbb{I} \times P_i$. For every ideal $\mathcal{I}$ in $\mathbb{I}$ and $D \in P_i$, $\mathcal{H}_{\mathcal{I},D} = \mathcal{I} \otimes \mathcal{D}_{i,D}$.

**7.6.4. Proposition:** Let $Y$ be a scheme and $\mathcal{E}$ a coherent sheaf on $X_2 \times Y$, flat over $Y$. Let $y \in Y$ be a closed point such that $\mathcal{E}_y \in \mathcal{I}(X_2)$. Then there exists an neighborhood $V \subset Y$ of $y$ such that

- $\mathcal{E}_{1|X \times V}$ and $\mathcal{E}_{|X \times V}$ are flat on $V$.

- For every closed point $v \in V$, $\mathcal{E}_v \in \mathcal{I}(X_2)$, and $(\mathcal{E}_1)_v = (\mathcal{E}_v)_1$, $(\mathcal{E}_{|X \times V})_v = (\mathcal{E}_v)_{|X}$.
[12] Drézet, J.-M. *Reachable sheaves on ribbons and deformations of moduli spaces of sheaves*. Intern. J. of Math. 28,12 (2017), 1750086.

[13] Drézet, J.-M. *Non-reduced moduli spaces of sheaves on multiple curves*. Advances in Geometry 20, 2 (2020), 285-296.

[14] Drézet, J.-M. *Primitive multiple schemes*. European J. of Math. 7 (2021), 985-1045.

[15] Drézet, J.-M. *Moduli of vector bundles on primitive multiple schemes*. International Journal of Mathematics, Vol. 34, No. 7 (2023) 2350035.

[16] Eisenbud, D., Green, M. *Clifford indices of ribbons*. Trans. of the Amer. Math. Soc., 1995, 347-3, 757-765.

[17] Ferrand, D. *Courbes gauches et fibrés de rang 2*. C.R. Acad. Sci. Paris, 281 (1977), 345-347.

[18] Gallego, F.J., González, M., Purnaprajna, B.P. *K3 double structures on Enriques surfaces and their smoothings*. J. of Pure and Applied Algebra 212 (2008), 981-993.

[19] González, M. *Smoothing of ribbons over curves*. J. reine angew. Math. 591 (2006), 201-235.

[20] Grothendieck, A. et al. SGA1. *Revêtements Étales et Groupe Fondamental*. SGA1. Lect. Notes in Math. 224. Springer-Verlag (1971).

[21] Hartshorne, R. *Stable reflexive sheaves*. Math. Ann. 254 (1980), 121-176.

[22] Savarese, M. *On the irreducible components of the compactified Jacobian of a ribbon*. https://arxiv.org/abs/1803.05360.

[23] Savarese, M. *Coherent Sheaves on Ribbons and their Moduli*. https://arxiv.org/abs/1902.08510.

[24] Savarese, M. *Generalized line bundles on primitive multiple curves and their moduli*. https://arxiv.org/abs/1902.09463.

INSTITUT DE MATHÉMATIQUES DE JUSSIEU - PARIS RIVE GAUCHE, CASE 247, 4 PLACE JUSSIEU, F-75252 PARIS, FRANCE

Email address: jean-marc.drezet@imj-prg.fr
*Proof.* This follows from theorem 6.2.1, 1-.

☐

The proof of the following result is similar of that of theorem 1.1.3 of [15], and shows that $\mathcal{I}(X_2)$ is a moduli space of sheaves in the sense of 2.6.

**7.6.5. Theorem:** Let $Y$ be a scheme and $\mathcal{E}$ a coherent sheaf on $X_2 \times Y$, flat over $Y$, such that for every closed point $y \in Y$, $\mathcal{E}_y \in \mathcal{I}(X_2)$. Then there exists a unique morphism $f_{\mathcal{E}} : Y \to \mathcal{I}(X_2)$ and an open cover $(Y_i)_{i \in I}$ such that for every $i \in I$, $f_{\mathcal{E}}^{\sharp}(\mathcal{H})|_{X_2 \times Y_i} \simeq \mathcal{E}|_{X_2 \times Y_i}$.

Let $\Gamma(X_2) \subset \mathrm{Pic}(X)$ be the algebraic subgroup of vector bundles that can be extended to $X_2$. According to 4.6, $\mathrm{Pic}(X_2)$ is an affine bundle on $\Gamma(X_2)$ with associated vector bundle $\mathcal{O}_{\Gamma(X_2)} \otimes \mathcal{H}^1(X, L)$. Let $p_X : X \times \Gamma(X_2) \to X$, be the projection. Then $\mathcal{I}(X_2)$ is an affine bundle on $X \times \Gamma(X_2)$, with associated vector bundle $(\mathcal{O}_{X \times \Gamma(X_2)} \otimes \mathcal{H}^1(X, L)) \oplus p_X^*(T_X \otimes L)$.

## REFERENCES

[1] Bangere, P., Mukherjee, J., Raychaudhry, D. *K3 carpets on minimal rational surfaces and their smoothing*. Internat. J. Math. 32 (2021), no. 6, Paper No. 2150032

[2] Bănică, C., Forster, O. *Multiple structures on space curves*. In: Sundararaman, D. (Ed.) Proc. of Lefschetz Centennial Conf. (10-14 Dec. Mexico), Contemporary Mathematics 58, AMS, 1986, 47-64.

[3] Bayer, D., Eisenbud, D. *Ribbons and their canonical embeddings*. Trans. of the Amer. Math. Soc., 1995, 347-3, 719-756.

[4] Chen, D., Kass, J. L. *Moduli of generalized line bundles on a ribbon*. J. of Pure and Applied Algebra 220 (2016), 822-844.

[5] Drézet, J.-M. *Faisceaux cohérents sur les courbes multiples*. Collect. Math. 2006, 57-2, 121-171.

[6] Drézet, J.-M. *Paramétrisation des courbes multiples primitives*. Adv. in Geom. 2007, 7, 559-612.

[7] Drézet, J.-M. *Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives*. Mathematische Nachrichten, 2009, 282-7, 919-952.

[8] Drézet, J.-M. *Sur les conditions d'existence des faisceaux semi-stables sur les courbes multiples primitives*. Pacific Journ. of Math. 2011, 249-2, 291-319.

[9] Drézet, J.-M. *Courbes multiples primitives et déformations de courbes lisses*. Annales de la Faculté des Sciences de Toulouse 22, 1 (2013), 133-154.

[10] Drézet, J.-M. *Fragmented deformations of primitive multiple curves*. Central European Journal of Mathematics 11, n° 12 (2013), 2106-2137.

[11] Drézet, J.-M. *Reducible deformations and smoothing of primitive multiple curves*. Manuscripta Mathematica 148 (2015), 447-469.

[12] Drézet, J.-M. *Reachable sheaves on ribbons and deformations of moduli spaces of sheaves*. Intern. J. of Math. 28, 12 (2017), 1750086.

[13] Drézet, J.-M. *Non-reduced moduli spaces of sheaves on multiple curves*. Advances in Geometry 20, 2 (2020), 285-296.

[14] Drézet, J.-M. *Primitive multiple schemes*. European J. of Math. 7 (2021), 985-1045.

[15] Drézet, J.-M. *Moduli of vector bundles on primitive multiple schemes*. International Journal of Mathematics, Vol. 34, No. 7 (2023) 2350035.

[16] Eisenbud, D., Green, M. *Clifford indices of ribbons*. Trans. of the Amer. Math. Soc., 1995, 347-3, 757-765.

[17] Ferrand, D. *Courbes gauches et fibrés de rang 2*. C.R. Acad. Sci. Paris, 281 (1977), 345-347.

[18] Gallego, F.J., González, M., Purnaprajna, B.P. *K3 double structures on Enriques surfaces and their smoothings*. J. of Pure and Applied Algebra 212 (2008), 981-993.
[19] González, M. *Smoothing of ribbons over curves*. J. reine angew. Math. 591 (2006), 201-235.

[20] Grothendieck, A. et al. SGA1. *Revêtements Etales et Groupe Fondamental*. SGA1. Lect. Notes in Math. 224. Springer-Verlag (1971).

[21] Hartshorne, R. *Stable reflexive sheaves*. Math. Ann. 254 (1980), 121-176.

[22] Savarese, M. *On the irreducible components of the compactified Jacobian of a ribbon*. https://arxiv.org/abs/1803.05360.

[23] Savarese, M. *Coherent Sheaves on Ribbons and their Moduli*. https://arxiv.org/abs/1902.08510.

[24] Savarese, M. *Generalized line bundles on primitive multiple curves and their moduli*. https://arxiv.org/abs/1902.09463.

INSTITUT DE MATHÉMATIQUES DE JUSSIEU - PARIS RIVE GAUCHE, CASE 247, 4 PLACE JUSSIEU, F-75252 PARIS, FRANCE

Email address: jean-marc.drezet@wanadoo.fr