Weighted point extractions and their exact first moments

Proof. On $M$, take $z=0$ as a local equation of $S$, and write the graph as $t=\varphi(x)$. The ideal of $C$ is $(z,t-\varphi(x))$. The blow-up chart containing $p$ is $t-\varphi(x)=zw$. The other chart contains the strict transform of $S$ and shows that $D\cap T$ avoids the singularity. The formula for $t$ proves the singularity assertion. The discrepancy of a primitive weighted blow-up of three regular parameters is the sum of its weights. Weighted adjunction gives the claimed pairs. For weights $(1,2,2)$, passing to the second Veronese identifies the coarse exceptional surface with $\mathbb P^2$; the generic index two along $x=0$ contributes the different $\Lambda/2$. It cannot be discarded in a stability calculation. ◻

Proof. Choose the ramification point at $[x:y]=[0:1]$ and its image at $[u:v]=[1:0]$. After scaling, write $$f=x^r y^{5-r}+\sum_{i>r}a_i x^i y^{5-i},\qquad g=y^5+\sum_{i\ge1}b_i x^i y^{5-i}.$$ Scaling $x$ with weight one and $v$ with weight $r$, and dividing the equation by its least weight, gives a flat family of centres whose central member is $$C_*:\quad u x^r y^{5-r}+v y^5=0.$$ For $r<5$, it is important to use this possibly nonreduced centre, rather than to cancel its factor $y^{5-r}$. Cancellation would change the Hilbert function and the expected order.

Here are the flatness and section-space details. The centres are relative Cartier divisors on the fixed smooth surface $S$, hence relative local complete intersections in $M$. Their ideals and all their powers have flat quotients over the parameter, as follows inductively from $I^a/I^{a+1}=\operatorname{Sym}^a(I/I^2)$. Thus their blow-ups form a flat family. The central blow-up $X_*$ is normal. Near the multiple component its only new local equations have the form $ab=s^{5-r}c$, or $ab=s^{5-r}$ with a unit suppressed. For $r=3,4$ these are canonical threefold hypersurfaces, and in particular are klt.

For $r=5$ the central centre is smooth and so is its blow-up. The divisor $B$ remains globally generated by the same ideal generators as in Lemma 90.1. The fibres over $\mathbb P^1$ are blow-ups of curvilinear length-five subschemes of the smooth conic. Their minimal resolutions are the successive blow-ups along that conic. A plane curve of degree $d$ distinct from the conic has sum of multiplicities along these clusters at most $2d$, by Bézout, and hence has anticanonical degree at least $d$. The conic itself has anticanonical degree one. The only curves of degree zero are the exceptional $(-2)$-chains, already contracted in $X_*$. Consequently $L$ is ample on each fibre. Since $L=T+B$ is nef, its morphism contracts no curve: a contracted curve would have $T$-degree zero and lie in a fibre. Hence $L$ is ample on $X_*$. Vanishing and base change identify the anticanonical section spaces of this flat family.

At the section $p$ of the family, the central blow-up is smooth and the action on its regular parameters has weights $(1,2,r-2)$. Choose the anticanonical linearization of weight zero at $p$. The weighted rescaling is precisely the local Rees construction for these parameters. Thus the central weight spaces identify with the actual associated graded section spaces at $G$. Indeed, a nonzero homogeneous central section restricts to a nonzero weighted homogeneous polynomial on this affine chart. It cannot have higher order than its weight. A weight basis lifted by base change therefore has independent initial forms and exhausts the original section space. This also rules out unaccounted cancellations.

We can now count the characters by a second, independent filtration, namely order along the strict transform $D_*$. Its defining section has weight zero: it is the quotient of the conic equation, of weight two, by the exceptional equation, also of weight two. For $0\le j\le m$, the $j$th quotient is the complete space $$H^0(D_*,\mathcal O(m+j,m+j)).$$ Completeness follows from vanishing for $(m+1)L-(j+1)D_*$, which is nef and big. Its dimension is $(m+j+1)^2$. In the chosen linearization, a monomial of target degree $i$ and source degree $k$ has weight $$(m-j)(r-2)+ri+k,\qquad 0\le i,k\le m+j.$$ The character shift $r-2$ comes from $E|_{D_*}=C_*$: the equation of $C_*$ has weight $r$, whereas the exceptional equation at $p$ has weight two. The mean weight of this quotient is therefore $$\frac{(3r-3)m+(5-r)j}{2}.$$

For $m<j\le\lfloor3m/2\rfloor$, the exact fixed factor is $E^{j-m}$. After removing it, sections are pulled back from $\mathcal O_M(2m,3m-2j)$. Restriction to the conic is surjective, so the quotient has dimension $$(2m+1)(6m-4j+1).$$ Its mean weight, including the fixed exceptional factor, is $$rm+\frac{6m-4j}{2}+2(j-m)=(r+1)m.$$ There are no further quotients. This is an exact finite-level description, not a count of an ambient space containing the sections. In particular their dimensions sum to $$h^0(X,mL)=\frac{10m^3+15m^2+11m+3}{3}.$$

Dividing the total character sum by $m h^0(X,mL)$ and taking the limit gives $$\begin{aligned} S_X(G) &=\frac3{10}\left[ \int_0^1(1+s)^2\, \frac{(3r-3)+(5-r)s}{2}\,ds +\int_1^{3/2}(12-8s)(r+1)\,ds\right]\\ &=\frac3{10}\,\frac{91r+25}{24} =\frac{91r+25}{80}. \end{aligned}$$ Subtracting from $A_X(G)=r+1$ proves the beta formula. No Zariski decomposition on the weighted extracted threefold has been computed or assumed. The weighted expected order is obtained from the actual graded sections. The divisor decompositions used to count them are on the central blow-up of $M$ and on its distinguished surface; these are separate from the weighted extraction at $p$. ◻