Smooth immersions, their conormal sequence, deformation charts and smooth sections

Local invariants at a point. Fix $x\in X$ and put $R={\mathcal{O}}_{Y,x}$, $S={\mathcal{O}}_{X,x}$, so that the closed immersion identifies $S$ with $R\mathrm{/}{\mathcal{I}}_x$ and $\kappa (x)$ with the residue field of both. Since $Y$ and $X$ are smooth of finite type over $k$ at $x$, [F1] provides standard smooth charts through $x$ and through $x$ in $X$; by [F2] the local rings $R$ and $S$ are regular local rings. Write $t={\mathrm{trdeg}}_k\kappa (x)$. Both $Y$ and $X$ are pure-dimensional and smooth, so their local scheme dimensions at $x$ equal $m$ and $n$; by [F3] applied to each, $\dim R=m-t$ and $\dim S=n-t$, hence $\dim R-\dim S=m-n=d$.

A regular sequence locally generating the ideal. By step 1.1 the local ring $R$ is regular and its quotient $S=R\mathrm{/}{\mathcal{I}}_x$ is regular, so [F4] applied to $I={\mathcal{I}}_x$ produces a regular system of parameters of $R$ whose first $d$ members $f_1,\dots ,f_d$ generate ${\mathcal{I}}_x$; in particular $f_1,\dots ,f_d$ is an $R$-regular sequence. Each germ has a representative on a common affine neighbourhood $U$ of $x$. That neighbourhood is Noetherian by [F2], so its ideal of definition of $U\cap X$ is finitely generated, and clearing denominators in the finitely many generation equalities shrinks $U$ until the representatives generate the ideal sheaf and their germs at $x$ remain the same regular sequence.

The conormal sheaf is free with basis the classes of the $f_j$. Let $S^d\to I\mathrm{/}{I}^2$ be the surjection sending the $j$-th basis vector to the class of $f_j$, which is onto by step 2.1. If ${\sum }_j{\bar{a}}_j[f_j]=0$ in $I\mathrm{/}{I}^2$, lift the ${\bar{a}}_j$ to $a_j\in R$; the relation says ${\sum }_j{a}_j{f}_j\in I^2$, say ${\sum }_j{a}_j{f}_j={\sum }_j{b}_j{f}_j$ with all $b_j\in I$, so ${\sum }_j(a_j-b_j)f_j=0$. By [F5] every $a_j-b_j$ lies in $I$, whence every ${\bar{a}}_j=0$. Thus $S^d\to I\mathrm{/}{I}^2$ is an isomorphism, and since $x$ was arbitrary $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ is locally free of rank $d$ with local basis the classes of $f_1,\dots ,f_d$.

Exactness of the conormal sequence and its dual. By [F6] the conormal map $\mathcal{I}\mathrm{/}{\mathcal{I}}^2\to i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}k}^1$ surjects onto the kernel $K$ of $i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}k}^1\to {\mathrm{\Omega }}_{X\mathrm{/}k}^1$. At $x$, [F7] makes $i^{\ast }{\mathrm{\Omega }}_{Y\mathrm{/}k,x}^1$ and ${\mathrm{\Omega }}_{X\mathrm{/}k,x}^1$ free over $S$ of ranks $m$ and $n$; the surjection onto the free target splits, so $K$ is free of rank $m-n=d$. By step 3.1 the conormal module is free of rank $d$, so the surjection between free modules of equal rank has nonzero determinant modulo the maximal ideal, hence is an isomorphism. Therefore the conormal sequence is short exact. Dualizing it at each point, the dual of a short exact sequence of finite free modules over a local ring is short exact, so $0\to T_X\to i^{\ast }{T}_Y\to N_{X\mathrm{/}Y}\to 0$ is exact with $N_{X\mathrm{/}Y}=(\mathcal{I}\mathrm{/}{\mathcal{I}}^2{)}^{\vee }$ locally free of rank $d$.

The étale-local coordinate model. Shrinking around $x$, choose $u_1,\dots ,u_n\in {\mathcal{O}}_Y(U)$ whose differentials on $X$ form a basis of ${\mathrm{\Omega }}_{X\mathrm{/}k}^1$ near $x$: differentials of local functions generate this sheaf, so select $n$ of them whose residues form a basis at $x$, then shrink until their determinant is a unit. Lift these functions through the surjection ${\mathcal{O}}_Y(U)\to {\mathcal{O}}_X(U\cap X)$. Form the $k$-morphism $\phi =({\phi }_1,\dots ,{\phi }_m):U\to {\mathbb{A}}_k^m$ with coordinates ${\phi }_j=f_j$ for $j\le d$ and ${\phi }_{d+j}=u_j$. By step 4.1 the classes of $d{f}_1,\dots ,d{f}_d$ form a basis of $I\mathrm{/}{I}^2\otimes \kappa (x)$ and the classes of $d{u}_1,\dots ,d{u}_n$ a basis of ${\mathrm{\Omega }}_{X\mathrm{/}k}^1\otimes \kappa (x)$; the exact conormal sequence of step 4.1 then shows that $d{\phi }_1,\dots ,d{\phi }_m$ span the $m$-dimensional $\kappa (x)$-vector space ${\mathrm{\Omega }}_{Y\mathrm{/}k,x}^1\otimes \kappa (x)$. Choosing a presentation of $U$ as an étale-locally standard smooth chart as in [F1] and adjoining the $m$ equations ${\phi }_j-t_j$ exhibits a standard smooth presentation of $U\to {\mathbb{A}}_k^m$ at $x$ whose Jacobian matrix has an invertible full $m\times m$ minor, because the corresponding row space together with the rows of the chart presentation spans. Hence $\phi$ is smooth of relative dimension $0$ at $x$ by [F1]. Its relative differentials vanish at $x$ by [F7], so $\phi$ is unramified and, being flat and locally of finite presentation by [F1], étale at $x$ by [F7]. Finally, ${\phi }^{-1}({\mathbb{A}}^n)\cap U$, where ${\mathbb{A}}^n\subset {\mathbb{A}}^m$ is the coordinate subspace $t_1=\cdots =t_d=0$, is exactly $U\cap X$, because $X$ is locally defined in $U$ by $f_1=\cdots =f_d=0$ by step 2.1.

The locally closed case. If $i:X\hookrightarrow Y$ is only a locally closed immersion, write it as a closed immersion $X\hookrightarrow U$ followed by an open immersion $U\hookrightarrow Y$; the ideal sheaf, the conormal sheaf and the conormal sequence are those of the closed part, computed in $U$, and the assertions are local near points of $X$, so they follow from the closed case applied to $X\hookrightarrow U$.

Smoothness of the deformation blowup. On the affine chart of ${\mathbb{P}}^1$ containing $\mathrm{\infty }$, use the coordinate $t$ vanishing at $\mathrm{\infty }$; the centre is cut out by $(f_1,\dots ,f_d,t)$. On its complement the centre is empty, so the blowup is the identity and is smooth. Near the centre the coordinate model is the blowup of ${\mathbb{A}}^m\times {\mathbb{A}}^1$ along $(f_1,\dots ,f_d,t)$. Its charts are computed from the Rees algebra: the $t$-chart has coordinates $(f_1\mathrm{/}t,\dots ,f_d\mathrm{/}t,t,u_1,\dots ,u_n)$ and the $f_j$-charts have coordinates $(f_j,f_l\mathrm{/}{f}_j(l\mathrm{\ne }j),t\mathrm{/}{f}_j,u_1,\dots ,u_n)$, each an affine space of dimension $m+1$; the chart ideals are generated by regular sequences, so each chart is smooth over $k$, and smoothness is local on the source, so the model blowup is smooth. Now let $\psi =\phi \times {\mathrm{id}}_{{\mathbb{P}}^1}:U\times {\mathbb{P}}^1\to {\mathbb{A}}^m\times {\mathbb{P}}^1$. It is étale, hence flat, and the ideal of $X\times \{\mathrm{\infty }\}$ in $U\times {\mathbb{P}}^1$ is the inverse image of the model ideal because $X\cap U={\phi }^{-1}({\mathbb{A}}^n)$. By [F9] (flat base change of blowups) the blowup of $U\times {\mathbb{P}}^1$ along $X\times \{\mathrm{\infty }\}$ is the base change of the smooth model blowup along $\psi$; by [F7] base changes of smooth morphisms are smooth, and a composition of smooth morphisms is smooth, so this base change is smooth over $k$. As $x\in X$ was arbitrary and $X$ is covered by the opens $U\cap X$, the blowup ${\mathrm{Bl}}_{X\times \{\mathrm{\infty }\}}(Y\times {\mathbb{P}}^1)$ is smooth over $k$.

Quasi-projectivity of the blowup. Assume now that $Y$ is quasi-projective over $k$. Then $Y\times {\mathbb{P}}_k^1$ is quasi-projective: $Y$ is a quasi-compact locally closed subscheme of some projective space, and the Segre embedding [F10] realizes the product of projective spaces as a closed subscheme of a projective space, so the product immerses quasi-compactly. Let $Z=Y\times {\mathbb{P}}_k^1$ and let $L$ be very ample on $Z$, hence ample [F10]. The ideal sheaf $\mathcal{J}$ of $X\times \{\mathrm{\infty }\}$ is coherent, so by [F10] the twist $\mathcal{J}\otimes L^{\otimes \nu }$ is globally generated for some $\nu \ge 1$; the graded algebra ${⨁}_{q\ge 0}{\mathcal{J}}^q\otimes L^{\nu q}$ has the same relative Proj as ${⨁}_{q\ge 0}{\mathcal{J}}^q$, since a trivialization of $L$ identifies their Proj charts and changing the trivialization rescales degree $q$ by the $q$-th power of a unit. A finite set of generators of $\mathcal{J}\otimes L^{\nu }$ therefore gives a graded surjection from ${\mathcal{O}}_Z[T_0,\dots ,T_r]$ onto this twisted Rees algebra. Its relative Proj is a closed subscheme of ${\mathbb{P}}_Z^r$, by the same homogeneous-quotient chart calculation as [F9], and a projective bundle over the quasi-projective $Z$ is quasi-projective, so the blowup is quasi-projective over $k$.

Sections of smooth morphisms. Let $s:T\to P$ be a section of a smooth morphism $P\to T$ of constant relative dimension $r$, with $T$ locally Noetherian. Near a point $t\in T$, [F8] factors $P\to T$ Zariski locally as an étale morphism $P^{\mathrm{\prime }}\to {\mathbb{A}}_T^r=\mathrm{Spec}{\mathcal{O}}_T[x_1,\dots ,x_r]$ followed by the projection, after shrinking around $s(t)$ and $t$; by [F7] the étale morphism is flat. The composite $T\to P^{\mathrm{\prime }}\to {\mathbb{A}}_T^r$ is the graph of the $T$-point with coordinates $c_i=s^{\ast }(x_i)$. This graph is cut out by the regular sequence $x_1-c_1,\dots ,x_r-c_r$: after quotienting by the first $i$ terms, the next element is a monic linear polynomial in a new variable and hence a nonzerodivisor over any base ring. Pulling the graph back along the flat étale map $P^{\mathrm{\prime }}\to {\mathbb{A}}_T^r$ gives a regular immersion $Q=T{\times }_{{\mathbb{A}}_T^r}{P}^{\mathrm{\prime }}\hookrightarrow P^{\mathrm{\prime }}$ of codimension $r$, and $Q\to T$ is étale. The section factors through a section $T\to Q$ of this étale morphism. By [F8], after shrinking near the chosen point, this section is described by a standard étale chart $\mathrm{Spec}((A[z]\mathrm{/}(P){)}_g)\to \mathrm{Spec}A$ with $z\mapsto a$, $P(a)=0$ and $P^{\mathrm{\prime }}(a)$ a unit. Factoring $P(z)=(z-a)Q_0(z)$ gives $Q_0(a)=P^{\mathrm{\prime }}(a)$; after restricting to the open neighborhood where $Q_0$ is invertible, the equation $P(z)=0$ forces $z=a$, so the section is an open immersion there. Hence near its image in $P^{\mathrm{\prime }}$, the pulled-back regular equations cut out $s(T)$, proving that $s$ is a regular immersion of codimension $r$. These equations remain regular after every base change $T^{\mathrm{\prime }}\to T$, since the section of the smooth base change $P{\times }_T{T}^{\mathrm{\prime }}\to T^{\mathrm{\prime }}$ is again locally obtained by this graph construction, including when $T^{\mathrm{\prime }}$ is singular. The normal bundle also commutes with base change: for a section of a smooth morphism, the conormal bundle identifies with $s^{\ast }{\mathrm{\Omega }}_{P\mathrm{/}T}^1$, and this relative differential bundle pulls back to $s^{\mathrm{\prime }\ast }{\mathrm{\Omega }}_{(P{\times }_T{T}^{\mathrm{\prime }})\mathrm{/}{T}^{\mathrm{\prime }}}^1$. Hence the codimension and normal bundle are preserved by arbitrary base change on $T$.