Point Blowup Resolution on Arbitrary Regular Surfaces

If $X=\mathrm{\varnothing }$ there is nothing to prove, so assume $X\mathrm{\ne }\mathrm{\varnothing }$. By [F1] fix a finite affine open cover $X=U_1\cup \cdots \cup U_r$ with $U_i=\mathrm{Spec}{A}_i$ and $A_i$ Noetherian; necessarily $r\ge 1$. This is a single existential instantiation of the cover granted by the definition of a Noetherian scheme, not a choice from an infinite family.

Fix $i$. The restriction $\mathcal{F}{\mathrm{\mid }}_{U_i}$ is quasi-coherent, and it is of finite type because $\mathcal{F}$ is coherent and the restriction of a coherent module to an open subscheme is coherent; by [F2] put $M_i:=\mathrm{\Gamma }(U_i,\mathcal{F})$, a finitely generated $A_i$-module. Each ${\mathcal{F}}_n{\mathrm{\mid }}_{U_i}$ is likewise a coherent, hence finite type, quasi-coherent subsheaf of $\mathcal{F}{\mathrm{\mid }}_{U_i}$, and $N_{i,n}:=\mathrm{\Gamma }(U_i,{\mathcal{F}}_n)$ is a finitely generated $A_i$-submodule of $M_i$.

For $m\le n$ the inclusion ${\mathcal{F}}_m\subseteq {\mathcal{F}}_n$ induces, under the equivalence of [F3], an inclusion of $A_i$-modules $N_{i,m}\subseteq N_{i,n}$; in particular $N_{i,1}\subseteq N_{i,2}\subseteq N_{i,3}\subseteq \cdots$ is an ascending sequence of submodules of $M_i$.

By [F4] the module $M_i$ is Noetherian, so the ascending sequence of step 3.1 stabilizes: there is an integer $s_i$ with $N_{i,n}=N_{i,s_i}$ for every $n\ge s_i$. This holds for each fixed $i$ with no choice used.

Set $s:=\max \{s_1,\dots ,s_r\}$, which exists because $r$ is finite. Let $n\ge s$. For every $i$ we have $N_{i,n}=N_{i,s}$, both being $N_{i,s_i}$, so the quasi-coherent subsheaves ${\mathcal{F}}_n{\mathrm{\mid }}_{U_i}$ and ${\mathcal{F}}_s{\mathrm{\mid }}_{U_i}$ of $\mathcal{F}{\mathrm{\mid }}_{U_i}$ have the same module of sections and are therefore equal by [F3]. Equality of subsheaves of a quasi-coherent sheaf can be checked on the members of an open cover, and the $U_i$ cover $X$; hence ${\mathcal{F}}_n={\mathcal{F}}_s$ for every $n\ge s$, which is the asserted stabilization with $n_0=s$.

Take a Noetherian affine neighbourhood $U=\mathrm{Spec}R$ of $p$, with center ideal $\mathfrak{n}$, and put $A=R_{\mathfrak{n}}={\mathcal{O}}_{S,p}$. The blowup restricts to the blowup on $U$ (Blowups restrict to open subschemes of the base). Flat base change along $R\to A$ identifies its further pullback with ${\mathrm{Bl}}_{\mathfrak{m}}\mathrm{Spec}A$, where $\mathfrak{m}=\mathfrak{n}A$ (Flat base change for blowups, and failure without flatness). The local ring of any point above $p$ is unchanged by this localization, because every element of $R\setminus \mathfrak{n}$ is already a unit there. Thus regularity above $p$ can be checked over $A$. Its dimension is one or two: $p$ cannot be a zero-dimensional irreducible component of the pure two-dimensional regular scheme, and the dimension is at most two. The two dimensions are treated separately below.

Suppose $\dim A=2$ and choose regular parameters $x,y$ generating $\mathfrak{m}$. The standard charts are $A[\mathfrak{m}\mathrm{/}x]$ and $A[\mathfrak{m}\mathrm{/}y]$ (Affine blowup standard charts and overlaps). By the power-torsion presentation of Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, the first is the quotient of $B^{\mathrm{\prime }}=A[T]\mathrm{/}(xT-y)$ by its $x$-power torsion. This torsion is zero. Indeed $A\mathrm{/}(x)$ is a regular one-dimensional local domain and $y$ is nonzero modulo $x$ by [F1]. If $xg=(xT-y)h$ in $A[T]$, reduction modulo $x$ gives $\bar{y}\bar{h}=0$ in the domain $(A\mathrm{/}(x))[T]$, so $h=x{h}_1$; cancelling $x$ in the domain $A[T]$ gives $g=(xT-y)h_1$. Thus multiplication by $x$ is injective on $B^{\mathrm{\prime }}$, and $A[\mathfrak{m}\mathrm{/}x]=B^{\mathrm{\prime }}$. It is a domain because it embeds into $A[1\mathrm{/}x]$. The other chart is symmetrically $A[U]\mathrm{/}(yU-x)$.

In the case $\dim A=1$, the one-dimensional regular local ring $A$ is a discrete valuation ring [F3], so $\mathfrak{m}$ is principal generated by a uniformizer, the center ideal is invertible on a neighbourhood of $p$, and the blowup of an invertible ideal is an isomorphism: $\pi$ is an isomorphism (Blowing up an effective Cartier divisor does nothing). Then $E={\pi }^{-1}(p)$ is the closed point $p$ with its reduced structure $\mathrm{Spec}\kappa (p)$, and it is an effective Cartier divisor.

Consider a prime $Q$ of $B=A[T]\mathrm{/}(xT-y)$ whose contraction $\mathfrak{q}$ in $A$ is not $\mathfrak{m}$. If $x\mathrm{\notin }\mathfrak{q}$, the localized chart is $B{\otimes }_A{A}_{\mathfrak{q}}=A_{\mathfrak{q}}$ by solving $T=y\mathrm{/}x$. If $x\in \mathfrak{q}$, then $y\mathrm{\notin }\mathfrak{q}$; in the localized chart $y=xT$ is a unit, so $x$ is a unit and the chart is $A_{\mathfrak{q}}[1\mathrm{/}x]$. In either case $B_Q$ is a localization of the regular ring $A_{\mathfrak{q}}$ and is regular. This includes contraction $\mathfrak{q}=(0)$, where the local ring is a field.

For a prime $Q$ of $B$ over $\mathfrak{m}$, its preimage $P$ in $C=A[T]$ is either $\mathfrak{m}C$ or $(x,y,h(T))$, where $h$ lifts a monic irreducible polynomial of $\kappa [T]$. The regular local ring $C_P$ has dimension two in the first case and three in the second: the prime chains $(0)⊊(x)⊊\mathfrak{m}C$ and, in the second case, $\mathfrak{m}C⊊P$ give the lower bounds, while the displayed two or three maximal-ideal generators give the upper bounds on dimension. Consequently $x,y$, or $x,y,h$, form a cotangent basis by [F1]. The class of $xT-y$ in $P{C}_P\mathrm{/}(P{C}_P{)}^2$ is the residue of $T$ times the class of $x$ minus the class of $y$, and is nonzero because the coefficient of $y$ is $-1$. The parameter-quotient lemma therefore makes $B_Q=C_P\mathrm{/}(xT-y)$ regular. Its dimension is one for $P=\mathfrak{m}C$ (the generic point of the exceptional curve), and two for $P=(x,y,h)$.

Steps 3.1 and 3.2 prove regularity on the $x$-chart, and the symmetric proof proves it on the $y$-chart; step 1.1 transfers these local rings to all points above $p$. Away from $p$ the blowup is an isomorphism (The blowup is an isomorphism off the center), so $S^{\mathrm{\prime }}$ is regular everywhere. It is pure of dimension two as well. In the dimension-one case this follows from the identity in step 2.2. In the dimension-two case the local charts above $p$ are domains of dimension two: all their prime-local dimensions are at most two, and step 3.2 exhibits dimension-two closed local rings. The exceptional divisor is not an irreducible component, since its chart equation $x$ (or $y$) is a nonzerodivisor. Thus every component meeting the exceptional divisor is the strict transform of the corresponding component through $p$ and has dimension two; every other component is unchanged. The components of the original regular surface all have dimension two, so the same holds for $S^{\mathrm{\prime }}$. The dimension-one local ring at the generic point of $E$ does not contradict pure dimension two.

The exceptional curve is effective Cartier: the pulled-back center ideal ${\mathcal{I}}_p{\mathcal{O}}_{S^{\mathrm{\prime }}}$ is invertible with nonzerodivisor local generators, and $E=V({\mathcal{I}}_p{\mathcal{O}}_{S^{\mathrm{\prime }}})$ is the associated effective Cartier divisor (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier). In the case $\dim A=2$, the normal cone theorem identifies $E$ with ${\mathrm{Proj}}_Z({\mathrm{gr}}_{{\mathcal{I}}_p}{\mathcal{O}}_S)$ over $Z=\{p\}$ (The exceptional divisor is the projectivized normal cone); on the one-point scheme $Z=\mathrm{Spec}\kappa (p)$ its graded pieces are ${\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}$, so $E\cong \mathrm{Proj}({\mathrm{gr}}_{\mathfrak{m}}A)$ and [F2] gives $E\cong \mathrm{Proj}\kappa (p)[X,Y]={\mathbb{P}}_{\kappa (p)}^1$ over $\kappa (p)$. In particular $E$ is a regular curve, so the blowup creates no new singular one-dimensional component.

Collecting: $S^{\mathrm{\prime }}$ is regular of pure dimension two and $E$ is effective Cartier by steps 4.1 and 5.1; in the two-dimensional case $E\cong {\mathbb{P}}_{\kappa (p)}^1$ by step 5.1; in the one-dimensional case $\pi$ is an isomorphism and $E=\mathrm{Spec}\kappa (p)$ by step 2.2; and $\pi$ restricts to an isomorphism off $E$ by step 4.1. Only regularity of $S$ and of the local charts was used, never smoothness over a field, so no smoothness claim is made.

Put $A={\mathcal{O}}_{X,p}$ with maximal ideal $\mathfrak{m}$, let $I=I_{Y,p}$ and $J=I_{Z,p}$ be the stalk ideals, so that $A\mathrm{/}I={\mathcal{O}}_{Y,p}$ is a one-dimensional regular local ring and hence a discrete valuation ring with valuation $v$ [F1]. Choose $x_1\in \mathfrak{m}$ whose image ${\bar{x}}_1\in A\mathrm{/}I$ is a uniformizer; then $\mathfrak{m}=I+(x_1)$, because both sides are ideals contained in $\mathfrak{m}$ and their images in the discrete valuation ring $A\mathrm{/}I$ are equal to its maximal ideal. The image $J(A\mathrm{/}I)$ is a nonzero ideal of the valuation ring, so $J(A\mathrm{/}I)=({\bar{x}}_1^N)$ for a unique integer $N\ge 1$, and $N=\min \{v(\bar{g}):g\in J\}$; pick $f\in J$ with $v(\bar{f})=N$. By [F2], $N={\mathrm{\ell }}_{A\mathrm{/}I}(A\mathrm{/}(I+J))=m_p(Y\cap Z)$. All claims are local over $p$ and its unique preimage, and blowups commute with the flat base change $\mathrm{Spec}A\to X$ (Flat base change for blowups, and failure without flatness), and localizing the chart saturation $(JB):x_1^{\mathrm{\infty }}$ commutes with taking this schematic closure: a fraction lies in the localized saturation exactly when some power of $x_1$ carries it into the localized ideal. Thus we compute in the local model ${\mathrm{Bl}}_{\mathfrak{m}}(\mathrm{Spec}A)$.

The center ideal on $Y$ is ${\mathcal{I}}_p{\mathcal{O}}_Y$, whose stalk at $p$ is $\mathfrak{m}(A\mathrm{/}I)=({\bar{x}}_1)$; it is not the zero ideal $I{\mathcal{O}}_Y$ defining $Y$ itself. It is the unit ideal away from $p$, and its generator at $p$ is a nonzerodivisor. Coherence allows the stalk generator to generate on a neighbourhood of $p$, so this ideal sheaf is invertible on $Y$. The strict transform of $Y$ is its blowup in this induced center ideal (Strict transforms of closed subschemes are blowups of the subscheme), and the blowup of an invertible ideal is an isomorphism (Blowing up an effective Cartier divisor does nothing). Hence $Y^{\mathrm{\prime }}\to Y$ is an isomorphism and has exactly one point $q$ above $p$, with local ring $A\mathrm{/}I$.

The $x_1$-chart of the local blowup is $B=A[\mathfrak{m}\mathrm{/}{x}_1]$. Mapping $y\mathrm{/}{x}_1^n\in B$ (with $y\in {\mathfrak{m}}^n$) to the unique element $\bar{a}\in A\mathrm{/}I$ with $\bar{y}=\bar{a}{\bar{x}}_1^n$ gives a well-defined $A$-algebra homomorphism $\psi :B\to A\mathrm{/}I$: existence holds because ${\mathfrak{m}}^n\subseteq I+(x_1^n)$, uniqueness because ${\bar{x}}_1$ is a nonzerodivisor of the domain $A\mathrm{/}I$, and compatibility with the relations of the localisation holds because $x_1^k(x_1^{m}y-x_1^n{y}^{\mathrm{\prime }})=0$ implies ${\bar{x}}_1^{k+m+n}(\psi (y\mathrm{/}{x}_1^n)-\psi (y^{\mathrm{\prime }}\mathrm{/}{x}_1^m))=0$. The composite of $\psi$ with the chart morphism $\mathrm{Spec}B\to \mathrm{Spec}A$ is the closed immersion $\mathrm{Spec}(A\mathrm{/}I)\hookrightarrow \mathrm{Spec}A$; since the pullback of $\mathfrak{m}$ to $\mathrm{Spec}(A\mathrm{/}I)$ is the invertible ideal $({\bar{x}}_1)$, the universal property of the blowup gives a unique lift $\mathrm{Spec}(A\mathrm{/}I)\to {\mathrm{Bl}}_{\mathfrak{m}}(\mathrm{Spec}A)$, and this lift is the strict transform $Y^{\mathrm{\prime }}$ in the chart (Universal property of the blowup, Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Strict transform of a closed subscheme).

In the chart the exceptional subscheme is $V(x_1)$: the pulled back center ideal $\mathfrak{m}\mathcal{O}$ is generated by the nonzerodivisor $x_1$ and $E=V(\mathfrak{m}\mathcal{O})$ (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains). Hence the scheme-theoretic intersection $Y^{\mathrm{\prime }}\cap E$ is computed in the chart by $A\mathrm{/}I{\otimes }_{B}B\mathrm{/}(x_1)=(A\mathrm{/}I)\mathrm{/}({\bar{x}}_1(A\mathrm{/}I))=A\mathrm{/}(I+(x_1))=A\mathrm{/}\mathfrak{m}=\kappa (p)$; so $Y^{\mathrm{\prime }}\cap E$ is a single reduced point $q$ with residue field $\kappa (p)$ and $m_q(Y^{\mathrm{\prime }}\cap E)={\mathrm{\ell }}_{{\mathcal{O}}_{X^{\mathrm{\prime }},q}}(\kappa (q))=1$, proving claim 2.

The strict transform of $Z$ in the chart is cut out by the saturation $J^{\mathrm{\prime }}=(JB):x_1^{\mathrm{\infty }}$, the closure of $V(JB)$ on the open chart complement of $E$ (Strict transform of a closed subscheme); it contains $f\mathrm{/}{x}_1$, because $x_1\cdot (f\mathrm{/}{x}_1)=f\in JB$. Let $K=\psi (J^{\mathrm{\prime }})\subseteq A\mathrm{/}I$ be the image, an ideal of the discrete valuation ring $A\mathrm{/}I$ containing $\psi (f\mathrm{/}{x}_1)=\bar{f}\mathrm{/}{\bar{x}}_1$, which has valuation $N-1$. If $q\in Z^{\mathrm{\prime }}$, then the closed subscheme $Y^{\mathrm{\prime }}\cap Z^{\mathrm{\prime }}$ of $Y^{\mathrm{\prime }}=\mathrm{Spec}(A\mathrm{/}I)$ contains the closed point $q$, so $K$ is a proper ideal contained in the maximal ideal $({\bar{x}}_1)$ and therefore $K=({\bar{x}}_1^j)$ for some $j\ge 1$; from ${\bar{x}}_1^{N-1}\in K$ we get $j\le N-1$, and hence $m_q(Y^{\mathrm{\prime }}\cap Z^{\mathrm{\prime }})={\mathrm{\ell }}_{A\mathrm{/}I}(A\mathrm{/}I\mathrm{/}K)=j\le N-1<N=m_p(Y\cap Z)$ [F1], proving claim 3.

Finally, if $m_p(Y\cap Z)=N=1$, then the element $\bar{f}\mathrm{/}{\bar{x}}_1$ of step 4.2 is a unit, so $K$ is the unit ideal whenever $q\in Z^{\mathrm{\prime }}$, a contradiction; hence $q\mathrm{\notin }{Z}^{\mathrm{\prime }}$, and since $q$ is the only point of $Y^{\mathrm{\prime }}$ over $p$ by step 2.1, the strict transforms $Y^{\mathrm{\prime }}$ and $Z^{\mathrm{\prime }}$ are disjoint over $p$. This completes claims 1-3 and the additional assertions.

Put ${\mathcal{I}}_p$ for the coherent ideal of the closed point. Multiplication gives a canonical graded surjection $\mathrm{Sym}({\mathcal{I}}_p)\twoheadrightarrow {⨁}_{n\ge 0}{\mathcal{I}}_p^n$: it sends a degree-$n$ product of local sections to their product in ${\mathcal{I}}_p^n$ (Symmetric algebra of a quasi-coherent module). Relative Proj is constructed on affine base opens (Relative Proj of a graded quasi-coherent algebra). On each standard homogeneous open, the quotient map induces a surjection on degree-zero localized coordinate rings, hence a closed immersion. These maps come from the same graded quotient and glue to the global closed immersion $Y_1\hookrightarrow {\mathrm{Proj}}_Y\mathrm{Sym}({\mathcal{I}}_p)$, the projective presentation asserted here. Locally the center has finitely many generators, so Blowups of finite type ideals are locally H-projective, and proper gives local embeddings into finite-dimensional projective spaces; it also gives global properness, and finite type follows from those local embeddings. Away from $p$ the blowup is an isomorphism (The blowup is an isomorphism off the center), so every fibre away from $p$ is a single point. By [F1], $A={\mathcal{O}}_{Y,p}$ is a Noetherian local domain of dimension one.

The fibre $E={\beta }^{-1}(p)$ is canonically ${\mathrm{Proj}}_Z({\mathrm{gr}}_{{\mathfrak{m}}_p}{\mathcal{O}}_Y)$ over $Z=\{p\}=\mathrm{Spec}\kappa (p)$ (The exceptional divisor is the projectivized normal cone); over the one-point base $Z$ this is $\mathrm{Proj}(S)$ with $S={\mathrm{gr}}_{\mathfrak{m}}A={⨁}_{n\ge 0}{\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}$, a graded $\kappa (p)$-algebra generated in degree one by the finite-dimensional space $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$. Fixing generators of $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ presents $S$ as a quotient of a polynomial ring $\kappa (p)[x_0,\dots ,x_r]$, so $E=V_{+}(I)=\mathrm{Proj}(S)$ is a closed subscheme of ${\mathbb{P}}_{\kappa (p)}^r$ with ${\mathcal{O}}_E(1)$ the restriction of the twisting sheaf ([F3]).

Isomorphism criterion. If $\beta$ is an isomorphism, then the pulled-back center ideal ${\mathfrak{m}}_p{\mathcal{O}}_{Y_1}$ is invertible (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier), and since an isomorphism identifies ideal sheaves and their invertibility, ${\mathfrak{m}}_p$ itself is invertible; then $\mathfrak{m}={\mathfrak{m}}_{p}A$ is generated by a nonzerodivisor, so ${\dim }_{\kappa }\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2=1=\dim A$ and $A$ is regular by the definition of regularity (embedding dimension and regular local ring). Conversely, if $A$ is regular, then $A$ is a discrete valuation ring by [F5] and ${\mathfrak{m}}_p$ is generated by a uniformizer, which is a nonzerodivisor and generates the maximal ideal at $p$ while ${\mathfrak{m}}_p$ is the unit ideal away from $p$; so ${\mathfrak{m}}_p$ is invertible and the blowup of an invertible ideal is an isomorphism (Blowing up an effective Cartier divisor does nothing), with ${\beta }_{\ast }{\mathcal{O}}_{Y_1}={\mathcal{O}}_Y$. This proves the equivalence and the final clause.

By [F2], $\chi (n)$ agrees eventually with a polynomial $an+b$ of degree one. Since $\chi (n)$ is nonnegative for every $n$ and $a\mathrm{\ne }0$, its leading coefficient $a$ is positive. For all sufficiently large $n$, the difference $\chi (n)-\chi (n-1)$ equals $a$; this difference is ${\mathrm{\ell }}_A({\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1})={\dim }_{\kappa (p)}{S}_n$, because the quotient is annihilated by $\mathfrak{m}$. Thus the graded Hilbert function is eventually a positive constant $c=a\in {\mathbb{Z}}_{>0}$.

Write $P=\kappa (p)[x_0,\dots ,x_r]$, $S=P\mathrm{/}I$, and $\mathfrak{b}=P_{+}$. The ideal sheaf ${\mathcal{I}}_E$ is coherent: on each standard affine chart it is a finite ideal in a Noetherian ring (Coherent module sheaves). By [F3], Serre vanishing applied to this ideal sheaf makes the global sections of $0\to {\mathcal{I}}_E(n)\to {\mathcal{O}}_{{\mathbb{P}}^r}(n)\to {\mathcal{O}}_E(n)\to 0$ exact on the right for $n\gg 0$. The sections of the middle sheaf are $P_n$, and its kernel is $(I^{\mathrm{s}\mathrm{a}\mathrm{t}}{)}_n$: a homogeneous polynomial represents a section of the ideal sheaf exactly when its fractions on every standard chart lie in the chart ideals, the saturation criterion of Closed subschemes of projective space and saturated ideals. Thus $H^0(E,{\mathcal{O}}_E(n))=P_n\mathrm{/}(I^{\mathrm{s}\mathrm{a}\mathrm{t}}{)}_n$ for large $n$. The polynomial ring $P$ is Noetherian (If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$), so the graded module $I^{\mathrm{s}\mathrm{a}\mathrm{t}}\mathrm{/}I$ has finitely many homogeneous generators. Each is annihilated by some power of $\mathfrak{b}$ by the definition of saturation; one common power ${\mathfrak{b}}^N$ annihilates them all. If their degrees are at most $D$, then in degree $n\ge D+N$ every coefficient multiplying such a generator has degree at least $N$ and lies in ${\mathfrak{b}}^N$. Hence $(I^{\mathrm{s}\mathrm{a}\mathrm{t}}\mathrm{/}I{)}_n=0$ for all such $n$. It follows that $H^0(E,{\mathcal{O}}_E(n))=S_n$ eventually. By step 3.1 its dimension is the positive constant $c$, so the Hilbert polynomial of ${\mathcal{O}}_E$ is the nonzero constant $c$. Its degree equals $\dim E$ by [F3], giving $\dim E=0$.

The space $E$ is a closed subscheme of the Noetherian space ${\mathbb{P}}_{\kappa (p)}^r$, hence is Noetherian of dimension zero by step 4.1, so it has finitely many points and all its local rings are zero-dimensional; as a closed subscheme of projective space $E$ is proper and of finite type over $\kappa (p)$, so $E$ is quasi-finite over $\kappa (p)$ and therefore finite over $\kappa (p)$ by [F4]. Combined with step 1.1, where the fibre over every $q\mathrm{\ne }p$ is a single point, every fibre of $\beta$ is finite with zero-dimensional local rings, so $\beta$ is quasi-finite; as $\beta$ is also proper by step 1.1, [F4] makes $\beta$ finite. This proves the first part of the statement and the finiteness of the fibre over $p$.

All assertions are proved: $\beta$ has the displayed closed immersion into ${\mathrm{Proj}}_Y\mathrm{Sym}({\mathcal{I}}_p)$, is locally H-projective, proper, finite type, finite, quasi-finite with fibre $\mathrm{Proj}({\mathrm{gr}}_{{\mathfrak{m}}_p}{\mathcal{O}}_{Y,p})$ over $p$, an isomorphism over $Y\setminus \{p\}$, and an isomorphism exactly when ${\mathcal{O}}_{Y,p}$ is regular, and the fiber over $p$ is finite by step 5.1.