1 Erratum to: Rend. Circ. Mat. Palermo, II. Ser DOI 10.1007/s12215-016-0276-4

The main result of the paper Proposition 1.3 is wrongly stated. Nevertheless, the proof of Proposition 1.3 and Proposition 1.5 provides a complete description of \({{{\text {Def}}}}_{\nu }({{\mathbf {k}}}[\epsilon ])\) and the paper needs only the corrections below.

2 Corrections

  • The statement of Proposition 1.3 has to be replaced by the following, which is exactly what is proved.

Proposition 1.3

Let \(\nu \,{:}\,X\hookrightarrow Y\) be a regular closed embedding of reduced algebraic schemes and let \( {{\text {Def}}}_{X/\nu /Y}\) be the deformation functor of \(\nu \) preserving X and Y (cf. [3, §3.4.1]). Then, there exists a surjective morphism \(\Phi \) from \({{\text {Def}}}_{\nu }({{\mathbf {k}}}[\epsilon ])\) to the fiber product

(4)

whose kernel is the image of the natural map .

Recalling that

figure a

by Proposition 1.5, we obtain the following result describing \({{{\text {Def}}}}_{\nu }(\mathbf{k}[\epsilon ])\), which is now to be considered the main result of the paper. (In the statement, the map is the one in the conormal sequence.)

Theorem

Let \(\nu \,{:}\,X\hookrightarrow Y\) be a regular closed embedding of reduced algebraic schemes. Then, there exists a long exact sequence

where the map \(\Theta \) is given by \(\Theta (\xi ,\eta )=\xi \circ \beta -\eta |_X\).

Proof

The second row of the above exact sequence follows from (the above version of) Proposition 1.3 and (\(\dagger \)).

By the definition of \({\text {Hom}}_{{\mathcal {O}}_X}(\nu ^*\Omega ^1_{Y},{\mathcal {O}}_{X})\) and \( {{{\text {Def}}}}_{\nu }(\mathbf{k}[\epsilon ])\) (cf. [3, p. 158 and p. 177]), an element mapped to zero by \(\Delta \) corresponds to a first order deformation

$$\begin{aligned} \tilde{\nu }:X\times {{\text {Spec}}}(\mathbf{k}[\epsilon ])\rightarrow Y\times {{\text {Spec}}}(\mathbf{k}[\epsilon ]) \end{aligned}$$

of \(\nu \) that is trivializable. More precisely, denoting by \(H_X\subset {{\text {Aut}}} ( X \times {\text {Spec}}( \mathbf{k}[\epsilon ]))\) the space of automorphisms restricting to the identity on the closed fiber and similarly for \(H_Y\subset {{\text {Aut}}} (Y \times {\text {Spec}}( \mathbf{k}[\epsilon ]))\), there exist \(\alpha \in H_X\) and \(\beta \in H_Y\), such that

$$\begin{aligned} \alpha \circ (\nu \times {{\text {Id}}}_{{\text {Spec}}(\mathbf{k}[\epsilon ])})\circ \beta =\tilde{\nu }. \end{aligned}$$

Then, one obtains a natural map \(H_X\times H_Y\rightarrow {{\text {Def}}}_{X/\nu /Y}({\mathbf {k}}[\epsilon ])\) whose image is the kernel of \(\Delta \). By (\(\dagger \)) and the well-known isomorphisms \(H_X\simeq {\text {Hom}}_{{\mathcal {O}}_X}(\Omega ^1_{X},{\mathcal {O}}_{X})\) and \(H_Y\simeq {\text {Hom}}_{{\mathcal {O}}_Y}(\Omega ^1_{Y},{\mathcal {O}}_{Y})\) (cf. [3, Lemma 1.2.6]), this map may be identified with \(\Theta \). The kernel of \(\Theta \) is by definition as in the statement. \(\square \)

  • In the first column of diagram (11), the vector space \( {{\text {Def}}}_{\nu }(\mathbf{k}[\epsilon ])\) must be replaced by the quotient \( {{\text {Def}}}_{\nu }(\mathbf{k}[\epsilon ])/{{\text {Im}}}( \Delta )\).

  • The paragraph “We remark that \({\text {Ext}}^1(\delta _1,\delta _0)\dots \) not isomorphic to it.” in §1.4 has to be replaced by the following:

“We remark that \({\text {Ext}}^1(\delta _1,\delta _0)\) coincides with \({{{\text {Def}}}}_{\nu }(\mathbf{k}[\epsilon ])\) in the case when \(f: X \rightarrow Y\) is a regular embedding. By (11), with \( {{\text {Def}}}_{\nu }(\mathbf{k}[\epsilon ])\) replaced by \( {{\text {Def}}}_{\nu }(\mathbf{k}[\epsilon ])/{{\text {Im}}} (\Delta )\), one has \(\varphi _1=\lambda -\mu \). Therefore,

$$\begin{aligned} {\text {Ext}}^1_{{\mathcal {O}}_X}(\Omega ^1_{X},{\mathcal {O}}_{X})\times _{{\text {Ext}}^1_{{\mathcal {O}}_X}(\Omega ^1_{Y}|_{X}, {\mathcal {O}}_{X})}{\text {Ext}}^1_{{\mathcal {O}}_Y}(\Omega ^1_{Y},{\mathcal {O}}_{Y})\simeq {{\text {Ker}}}(\varphi _1), \end{aligned}$$

\(\Delta \) coincides with \(\partial \) and \(\Theta \) with \(\varphi _0\). Example 1.7 below gives an instance where \(\partial =\Delta \) is nonzero.”

  • In the proof of Lemma 2.1, the exact sequence (13) is not exact on the left, but this does not affect the proof.

  • Replace the statement of Corollary 2.2 by the following:

Corollary 2.2

There is a natural surjective map

$$\begin{aligned} \tau : {{\text {T}}}_{(S, C)}{\mathcal {V}}_{m,\delta }\longrightarrow {{\text {Def}}}_{\phi }(\mathbf{k[\epsilon ]})\simeq {{\text {Def}}}_{\nu }(\mathbf{k[\epsilon ]}). \end{aligned}$$

Moreover, if X is stable, then the differential of the moduli map of \(\psi _{m,\delta }\) at (S, C) factors as

$$\begin{aligned}&d_{(S,C)}\psi _{m,\delta }: {{\text {T}}}_{(S, C)}{\mathcal {V}}_{m,\delta }\mathop {\longrightarrow }\limits ^{\tau }{{\text {Def}}}_{\nu }(\mathbf{k[\epsilon ]}) \\&\quad \longrightarrow {{\text {Def}}}_{\nu }(\mathbf{k[\epsilon ]})/ {{\text {Im}}}(\Delta ) \mathop {\longrightarrow }\limits ^{ p _X}{{\text {Ext}}}^1_{{\mathcal {O}}_X}(\Omega _{X},{\mathcal {O}}_{X})\simeq T_{[X]}{\overline{\mathcal M}}_g, \end{aligned}$$

where \(p_X\) is the map appearing in the correct version of (11).

In particular, if \({{\text {Ext}}}^2_{{\mathcal {O}}_Y}(\Omega ^ 1_{Y}(X),{\mathcal {O}}_{Y})=0\), then \(d_{(S,C)}\psi _{m,\delta }\) is surjective; if

$$\begin{aligned} {{\text {Ext}}}^1_{{\mathcal {O}}_Y}\left( \Omega ^ 1_{Y}(X),{\mathcal {O}}_{Y}\right) ={{\text {Hom}}}_{{\mathcal {O}}_X}\left( \Omega ^ 1_{Y}|_X,{\mathcal {O}}_{X}\right) = {{\text {Hom}}}_{{\mathcal {O}}_Y}\left( \Omega ^ 1_{Y},{\mathcal {O}}_{Y}\right) =0 \end{aligned}$$

then \(d_{(S,C)}\psi _{m,\delta }\) is injective.

  • At the end of Remark 2.3, add “In this case, using the above notation, one has \( {{\text {Hom}}}_{{\mathcal {O}}_Y}(\Omega ^ 1_{Y},{\mathcal {O}}_{Y})=H^ 0(Y,T_Y)=0\), and moreover, by [2, (4) in proof of Prop. 1.2], \({{\text {Hom}}}_{{\mathcal {O}}_X}(\Omega ^ 1_{Y}|_X,{\mathcal {O}}_{X})= H^0(X, {T_Y}_{|_X})=0\).”