Blowing up a non-regular point strictly increases the finite normalization subalgebra

By [F1] the inclusion ${\mathcal{O}}_Y\subseteq {\beta }_{\ast }{\mathcal{O}}_{Y_1}\subseteq {\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$ holds as ${\mathcal{O}}_Y$-algebras, and ${\beta }_{\ast }{\mathcal{O}}_{Y_1}$ is coherent. If the first inclusion were an equality, then the finite morphism $\beta$ would satisfy ${\beta }_{\ast }{\mathcal{O}}_{Y_1}={\mathcal{O}}_Y$; on an affine chart $U=\mathrm{Spec}R\subseteq Y$ with ${\beta }^{-1}(U)=\mathrm{Spec}B$ this says that the image of the structure map $R\to B$ generates $B$ as an $R$-module, so $B=R$ and $\beta {\mathrm{\mid }}_{{\beta }^{-1}(U)}$ is an isomorphism; hence $\beta$ would be an isomorphism. But $p$ is not a regular point, so $\beta$ is not an isomorphism by [F2]. Therefore ${\mathcal{O}}_Y⊊{\beta }_{\ast }{\mathcal{O}}_{Y_1}$: the inclusion is strict.

The quotient $\mathcal{Q}={\beta }_{\ast }{\mathcal{O}}_{Y_1}\mathrm{/}{\mathcal{O}}_Y$ is coherent by [F3] and is nonzero by step 1.1. Away from $p$ the morphism $\beta$ is an isomorphism, so $({\beta }_{\ast }{\mathcal{O}}_{Y_1}{)}_q={\mathcal{O}}_{Y,q}$ for every $q\mathrm{\ne }p$, and the stalk ${\mathcal{Q}}_q=0$ there; hence the support of $\mathcal{Q}$ is contained in the closed point $p$, and therefore equals $\{p\}$. By [F3] the stalk ${\mathcal{Q}}_p$ has finite length over ${\mathcal{O}}_{Y,p}$.

This proves the first assertion. For the general step, argue by induction on $i$. At each stage $Y_{i-1}$ is an integral Noetherian one-dimensional scheme with a fixed finite normalization ${\nu }_{i-1}:Y^{\nu }\to Y_{i-1}$: for $i-1=0$ this is $\nu$; inductively, $Y_i$ is the blowup of the integral scheme $Y_{i-1}$ in a nonzero ideal, hence is integral (Blowing up a nonzero ideal on an integral scheme is birational) and Noetherian, and The finite normalization of a curve factors through the blowup of a closed point shows that $Y^{\nu }$ is a normalization of $Y_i$ as well. Let ${\beta }_i:Y_i={\mathrm{Bl}}_{p_{i-1}}{Y}_{i-1}\to Y_{i-1}$ be the blowup at a closed non-regular point $p_{i-1}$, so that by the already proved first assertion applied to $Y_{i-1}$ and $p_{i-1}$ the sequence of ${\mathcal{O}}_{Y_{i-1}}$-modules $0\to {\mathcal{O}}_{Y_{i-1}}\to ({\beta }_i{)}_{\ast }{\mathcal{O}}_{Y_i}\to {\mathcal{Q}}_i\to 0$ is exact with ${\mathcal{Q}}_i\mathrm{\ne }0$.

Push the short exact sequence of step 3.1 forward along the finite affine morphism $f_{i-1}$. By [F4] this gives $0\to f_{i-1,\ast }{\mathcal{O}}_{Y_{i-1}}\to f_{i,\ast }{\mathcal{O}}_{Y_i}\to f_{i-1,\ast }{\mathcal{Q}}_i\to 0$. This last sheaf is nonzero: choose an affine open $V=\mathrm{Spec}R$ of $Y$ containing the image of the center. Its inverse image is affine, say $\mathrm{Spec}B$, and ${\mathcal{Q}}_i$ restricts there to a nonzero finite $B$-module $M$, because its nonzero center stalk lies on that open. The pushforward restricts to the same nonzero module $M$ viewed as an $R$-module. Restriction of scalars is faithful, so $f_{i-1,\ast }{\mathcal{Q}}_i\mathrm{\ne }0$. Consequently the pushed-forward inclusion is strict. Both terms embed as coherent subalgebras in ${\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$ by the normalization factorization at each stage. No assertion that ${\mathcal{Q}}_i$ is a sheaf on the reduced residue-field point is needed: its stalk may have nontrivial nilpotent maximal-ideal action.

Collecting: the inclusion ${\mathcal{O}}_Y⊊{\beta }_{\ast }{\mathcal{O}}_{Y_1}$ is strict with quotient a nonzero coherent sheaf of finite length supported exactly at $p$, and every further point blowup at a closed non-regular center strictly increases the pushed-forward structure sheaf inside the fixed finite normalization ${\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$.