Nonsquare tangent-conic surface singularities terminate under point blowups

Choose generators $x_1,x_2,x_3$ of $\mathfrak{m}$ with ${\mathrm{gr}}_{\mathfrak{m}}A=\kappa [T_1,T_2,T_3]\mathrm{/}(q)$, $q$ a nonzero quadratic, and lift the unique quadratic relation to $\sum a_{ij}{x}_i{x}_j=\sum a_{ijk}{x}_i{x}_j{x}_k$ modulo terms of order at least four; on the $x_1$-chart, with $y_i=x_i\mathrm{/}{x}_1$, dividing by $x_1^2$ exhibits the exceptional conic as $q(1,y_2,y_3)=0$.

The rational-domain definition puts $A$ essentially of finite type over a field or a complete equicharacteristic local base. In positive characteristic this forces both $A$ and its residue field to have that prime characteristic. In characteristic zero the base contains $\mathbb{Q}$: for a complete equicharacteristic local base every nonzero integer has nonzero residue and is a unit. Its images remain units in $A$ and every residue field. Hence $A$ is equicharacteristic, and the fixed-coordinate formal-arc supplier applies; this premise is derived from the rationality class, not from normal completion. Each singular successor is a normal two-dimensional local ring essentially of finite type over $A$, with the same fraction field. Rationality propagates by the stated modification supplier; it remains in the same base class, so the regular-fibre normality supplier gives its normal completion.

At a closed point $p$ of that conic, if the fibre equation were not in the square of the plane maximal ideal, then the local maximal ideal of the surface would be generated by $x_1$ and two lifts and have embedding dimension at most two, making the local ring regular; so every singular point has the fibre quadratic in $I_p^2$, and if the residue degree of $p$ exceeded one, the quadratic square-ideal lemma would make $q$ a scalar times a square, contrary to hypothesis; hence every singular point is $\kappa$-rational.

Changing the triple so that $p=(1,0,0)$, the original conic equation is a nonsquare binary quadratic in the two coordinates transverse to $p$: if it has a $\kappa$-root it splits into two distinct $\kappa$-lines meeting at the unique singular point of the conic, and otherwise the conic has only the rational vertex; in either case there is at most one singular successor.

Absorbing into the cubic part the coefficients of the three monomials $x_1{x}_i$ with zero residue, the chart relation forces the remaining cubic coefficient $a_{111}$ to lie in $\mathfrak{m}$, since otherwise it would express $x_1$ in the square of the chart maximal ideal and eliminate it as a cotangent generator; writing that coefficient modulo $(x_2,x_3)$ as $b{x}_1$, the successor quadratic relation restricts on the plane $x_1=0$ to exactly the same nonsquare binary quadratic, with all other terms carrying a factor of $x_1$; hence a singular successor again has nonsquare tangent conic and a singular successor again has $\kappa$-rational singular points by the argument of step 2.1. On $x_1=0$, a $\kappa$-root of the nonsquare binary quadratic is simple (and if it has no $\kappa$-root there is no such point), so the local fibre equation is not in the square of the plane maximal ideal. Such a point is regular by step 2.1. Thus every singular successor point lies off $x_1=0$, justifying continued use of the same coordinate chart.

If an infinite chain of singular point blowups existed, its centres would be $\kappa$-rational with residue field $\kappa$ and the fixed element $x_1$ would generate the pullback of every centre ideal, so the fixed-coordinate helper would construct a nonsingular formal arc, namely a surjection from the completion of $A$ onto a complete discrete valuation ring with uniformizer the image of $x_1$ whose point-blowup centres are the given ones; $\kappa$-rationality and rationality of the local rings propagate along the chain, and normal completion persists by the regular-fibre normality result.

The completion $\widehat{A}$ is a normal Noetherian local domain of dimension two. Applying the formal-arc termination helper to its surjection onto the complete DVR makes the point blowups of $\widehat{A}$ regular at the arc centre after finitely many steps.

Blowup flat base change identifies the completed blowups with the blowups of the completion, and completion base change preserves the closed-fibre local completions; regularity is preserved and reflected by completion, so those finitely many completed steps give a regular local ring on the original chain, contradicting the existence of infinitely many singular successors; the chain of singular blowups is therefore finite and ends in a regular model.

Thus each blowup has at most one singular successor, that successor is residue-field rational and again has nonsquare tangent conic, and the Axiom of Choice and the Axiom of Dependent Choice are inherited from the cited suppliers; no square-conic case is asserted.