The blowup of a one-dimensional integral Noetherian scheme at a closed point is finite

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.