Section 86

The actual refined sections and their expected orders

Part XII · Family 3.2: double conics in a projective bundlePDF page 300

For the exceptional divisor $G$ , put $V_m^{\ge j}=\{s\in H^0(X,-mK_X):\operatorname{ord}_G(s)\ge j\}, \qquad W_{m,j}=V_m^{\ge j}/V_m^{\ge j+1}.$ The spaces $W_{m,j}$ are viewed as sections on $G$ by taking initial terms. Their basis divisors are normalized by $m h_m$ , where $h_m=h^0(X,-mK_X)$ , and summed over all primary weights $j$ . Thus, for a prime divisor $E$ over $G$ , the expected order $S(W;E)$ is the limit of the maximal normalized order sum of bases of these spaces. We use the refined threshold $\delta(G,\Gamma/2;W)= \inf_E\frac{A_{G,\Gamma/2}(E)}{S(W;E)}.$ This is the multigraded version of the definition in Section 2.

On the coarse surface $G$ , the coordinate $z$ is a rational section with zero divisor $\Gamma/2$ . Its square has bidegree $(2,1)$ . Consequently the primary piece of weight $j$ has rational divisor class $(m,j/2)$ . Writing $j=2n+\varepsilon$ leaves a polynomial part of bidegree $(m-\varepsilon,n)$ and the fixed factor $z^\varepsilon$ . The contribution of this factor to an asymptotic basis divisor is bounded by $O(1/m)$ for each fixed valuation. The different $\Gamma/2$ is a separate, fixed boundary and is never discarded.

Lemma 86.1 (The actual primary filtration). Let $G$ be the weighted exceptional divisor of Lemma 84.4. Its filtration on the homogeneous coordinate quotient has weights $\operatorname{wt}(x_0,x_1,y_0,y_1,u_0,r,z)=(0,2,0,2,0,0,1).$ The associated graded ring is the quotient by $F_0$ in Lemma 84.2. In particular, its weights are the actual orders along $G$ .

Proof. The initial equation is irreducible by Lemma 84.2. The associated graded ring for the indicated weight filtration is its initial quotient and is a domain. Hence the induced order is multiplicative. The initial domain contains neither $x_0$ nor $y_0$ in its defining prime ideal, so localization at $x_0y_0$ introduces no additional initial relations. On $x_0=y_0=1$ , it has orders $1,2,2$ on $z,a,b$ and agrees with the divisorial order of the weighted extraction.

We also check that the homogeneous calculation computes all anticanonical sections. Write $Q$ for the projective bundle, and retain the degrees $(2,3,2)$ of $X$ and $(2,1,1)$ of $-K_X$ . For $m\ge2$ , the direct image of $\mathcal O_Q(2m-2,m-3,m-2)$ is a sum of line bundles on $\mathbb P^1\times\mathbb P^1$ of degrees $(2m-2-j-k,m-3-j-k),\qquad j+k\le m-2.$ The first degree is nonnegative and the second is at least $-1$ . Hence their first cohomology vanishes. For $m=1$ , the relative degree is $-1$ , and all relative cohomology vanishes. The restriction sequence consequently identifies $H^0(X,-mK_X)$ with the degree $(2m,m,m)$ part of the homogeneous quotient. This argument applies equally to the initial hypersurface, so no additional sections or cancellations are omitted. ◻

Lemma 86.2 (The refined surface series). Put $s=u_0$ , $a=x_1$ and $b=y_1$ on the chart $x_0=y_0=1$ . Thus the initial relation is $z^2+af+bg=0$ , where $f=s(cr+ds)$ and $g=r^2+e rs+h s^2$ . The exceptional surface is $G=\mathbb P^1_{[r:s]}\times\mathbb P^1_{[a:b]}$ . After removing the factor $z^\varepsilon$ , the piece of primary weight $2n+\varepsilon$ , where $\varepsilon\in\{0,1\}$ , is exactly $\bigoplus_{k=0}^{n}a^{n-k}b^k s^{M_{m,n,k,\varepsilon}} \operatorname{Sym}^{m-\varepsilon-M_{m,n,k,\varepsilon}}\langle r,s\rangle, \qquad M_{m,n,k,\varepsilon}= \max\{0,n-m-k,2k-m+\varepsilon\}.$ A summand with negative exponent is zero. Equivalently, the whole refined series has a monomial basis indexed by $0\le v\le m-\varepsilon,\qquad 0\le p\le m+v,\qquad 0\le k\le\left\lfloor\frac{m-\varepsilon+v}{2}\right\rfloor,$ whose element is $z^\varepsilon s^v r^{m-\varepsilon-v}a^pb^k$ and whose primary weight is $2(p+k)+\varepsilon$ .

Proof. Before using the relation, a spanning monomial is $s^i r^j z^{2h+\varepsilon}a^p b^q, \qquad i+j+2h+\varepsilon=m,\quad 0\le p\le m+i,\quad 0\le q\le i.$ Set $n=p+q+h$ and expand $z^{2h}=(-af-bg)^h$ . Suppose that $\ell$ of the factors contribute $af$ . The resulting exponent of $b$ is $k=q+h-\ell$ , and its binary coefficient is divisible by $s^{i+\ell}$ . We have $n-m-k=p+\ell-m\le i+\ell,$ and $2k-m+\varepsilon =2q+2h-2\ell-m+\varepsilon =2q-i-j-2\ell\le i+\ell.$ Thus every actual piece is contained in the displayed direct sum. The second description is simply the set of monomials in that direct sum.

These inclusions are equalities by an exact dimension count. Put $N=m+v$ . For each $v$ , the even piece contributes $(N+1)(\lfloor N/2\rfloor+1)$ monomials; for $v\le m-1$ , the odd piece contributes $(N+1)(\lfloor(N-1)/2\rfloor+1)$ . Pairing the two contributions gives $(N+1)^2$ . The unpaired even contribution with $v=m$ is $(2m+1)(m+1)$ . Hence the total dimension of the containing spaces is $\sum_{N=m}^{2m-1}(N+1)^2+(2m+1)(m+1) =\frac{(2m+1)(7m^2+7m+6)}6.$ This is $h^0(X,-mK_X)$ , by Riemann–Roch and $(-K_X)^3=14$ . The actual associated graded pieces have this same total dimension. Every inclusion is therefore an equality. In particular, the description remains valid when $f$ or $g$ has a double root. ◻

Remark 86.3 (The ample-series condition). For $n=m$ and $\varepsilon=0$ , the exact description contains $a^{\lceil m/2\rceil} H^0\bigl(G,\mathcal O_G(m,\lfloor m/2\rfloor)\bigr).$ Indeed, its exponent $k$ of $b$ lies between zero and $\lfloor m/2\rfloor$ , and then $M_{m,m,k,0}=0$ . Along even multiples this is a complete series of an ample divisor, multiplied by a fixed section. Thus the refined graded series contains an ample series, as required in Abban–Zhuang adjunction.

Lemma 86.4 (The expected primary order). For the weighted extraction above a double conic, $A_X(G)=3,\qquad S_X(G)=\frac{135}{56},\qquad A_X(G)-S_X(G)=\frac{33}{56}.$

Proof. The discrepancy is the sum $1+2$ of the transverse weights. Let $T_m$ be the sum of the primary weights of an adapted basis. For fixed $v$ and $\varepsilon$ , put $N=m+v$ and $K=\lfloor(N-\varepsilon)/2\rfloor$ . The contribution is $\sum_{p=0}^{N}\sum_{k=0}^{K}\bigl(2p+2k+\varepsilon\bigr) =(N+1)(K+1)(N+K+\varepsilon).$ For $m\le N\le2m-1$ , pairing the even and odd contributions gives $3N(N+1)^2/2$ , irrespective of the parity of $N$ . The remaining contribution, with $N=2m$ and $\varepsilon=0$ , is $3m(2m+1)(m+1)$ . Therefore $\begin{aligned} T_m&=\frac32\sum_{N=m}^{2m-1}N(N+1)^2+3m(2m+1)(m+1)\\ &=\frac{m(45m^3+62m^2+63m+22)}8. \end{aligned}$ Since $h^0(X,-mK_X)=(2m+1)(7m^2+7m+6)/6$ , taking the limit of $T_m/(mh^0(X,-mK_X))$ proves the assertion. ◻

Example 86.5. For $m=1$ , the primary pieces of weights $0,1,\ldots,6$ have dimensions $2,\ 1,\ 3,\ 1,\ 2,\ 0,\ 1.$ The even pieces are $\langle r,s\rangle,\quad \langle ar,as,bs\rangle,\quad \langle a^2s,abs\rangle,\quad \langle a^2bs\rangle,$ and the odd pieces are $\langle z\rangle$ and $\langle az\rangle$ . Their dimensions sum to $10$ and their weights sum to $24$ .

Remark 86.6. Ignoring bounded parity shifts, the basis is parametrized by the rational polytope $\{(x,y,w):0\le x\le1,\ 0\le y\le1+x,\ 0\le w\le(1+x)/2\}.$ Here $x=v/m$ , $y=p/m$ and $w=k/m$ . Its volume is $7/6$ , and its barycentre is $\left(\frac{17}{28},\frac{45}{56},\frac{45}{112}\right).$ Consequently, the refined expected orders along $s=0$ , $r=0$ , $a=0$ and $b=0$ are, respectively, $\frac{17}{28},\qquad\frac{11}{28},\qquad \frac{45}{56},\qquad\frac{45}{112}.$ The odd factor $z$ changes normalized expectations by $O(1/m)$ for a fixed valuation. This does not remove the different $\frac12\Gamma$ , where $\Gamma=(af+bg=0)$ : the different is part of the fixed log pair on $G$ . Neither the actual series nor its primary moment has been obtained from a Zariski decomposition on the weighted threefold extraction.

Lemma 86.7 (An exact secondary trace). Suppose that the initial equation is $z^2+as^2+br^2=0$ . Give $(r,s,a,b,z)$ secondary weights $(1,0,2,0,1)$ . The sum of the secondary weights of a basis of the complete primary associated graded space is exactly $2m\,h^0(X,-mK_X).$ Thus the secondary expected order is $2$ , with no finite-level error.

Proof. The equation is homogeneous of secondary weight $2$ . In the monomial basis of Lemma 86.2, the secondary weight of $z^\varepsilon s^v r^{m-\varepsilon-v}a^pb^k$ is $m-v+2p$ . For fixed $v,k,\varepsilon$ , the index $p$ runs from $0$ to $m+v$ . Its average secondary weight is therefore $m-v+(m+v)=2m.$ Summing over the other indices gives the identity. ◻