Surface calculations and their normalization

The refinement theorem concerns spaces of sections, including their fixed divisors. A numerical equivalence of divisors does not identify such spaces. This distinction matters when a restriction map is not onto, when points become infinitely near, and when a surface is replaced by its resolution. We give the formulas used below in a form which records these issues. They are the surface calculations underlying [3, Section 1.7].

4.1 A family of complete surface series

Let $L$ be ample on a threefold $X$, put $V=L^3$, and let $W_{m,j}$ be an actual refinement of $H^0(X,mL)$. Thus $N_m=\sum_j\dim W_{m,j}=Vm^3/6+O(m^2)$. Suppose that on a smooth projective surface $T$, after pulling back the sections, its pieces are complete moving series with their fixed factors: $$W_{m,j}=s_{mJ_t}H^0(T,mP_t),\qquad t=j/m.$$ Here $P_t$ is nef, $J_t$ is an actual effective divisor, and both are rational on rational parameters. The parameter belongs to a bounded union of intervals. Divisibility and bounded rounding are understood. We require the uniform section estimates used in taking the following Riemann sums; they hold in our applications because the divisors range over finitely many rational polyhedral chambers with the section descriptions proved in the corresponding part. It also suffices to have inclusions in the displayed spaces whose total dimension deficit is $o(m^3)$, with linearly bounded orders, by Lemma 2.5. Equal total leading dimensions are useful only after those inclusions have been proved.

For a big nef divisor $P$, write $S_P(v)=(P^2)^{-1}\int_0^\infty\mathop{\mathrm{vol}}(P-xv)\,dx$, with the usual birational interpretation of the integrand. Products $P_t^2S_{P_t}(v)$ at zero-volume endpoints are interpreted by limits. Then (4.1)$$\label{az:eq:surface-family} \frac3V\int P_t^2\,dt=1,\qquad S(W;v)=\frac3V\int P_t^2\bigl(v(J_t)+S_{P_t}(v)\bigr)\,dt.$$ Indeed, a piece has $m^2P_t^2/2+o(m^2)$ sections. Its fixed factor adds $mv(J_t)$ to each order, while its complete moving series has mean order $mS_{P_t}(v)+o(m)$. There are order $m$ pieces; dividing their sums by $mN_m$ gives the coefficient $3/V$. This proof also shows why a fixed divisor cannot simply be discarded.

In particular, suppose at a chosen point of $T$ that $S_{P_t}(v)\le b(t)A_{T,\Delta}(v)$ and $v(J_t)\le e(t)A_{T,\Delta}(v)$ for every valuation under consideration. Then (4.2)$$\label{az:eq:surface-average-bound} S(W;v)\le\frac3V\left(\int P_t^2(b(t)+e(t))\,dt\right) A_{T,\Delta}(v).$$ Thus the relevant bound is the dimension-weighted average of the surface bounds. It need not equal their worst value. If the spaces only embed in complete spaces and the leading dimensions differ, the same argument gives an upper bound with the original $N_m$; one must not normalize by the larger dimension.

4.2 A curve on the surface and a point on that curve

Suppose first that $C\subset T$ is a smooth rational curve. Put $c_t=\mathop{\mathrm{ord}}_CJ_t$ and take the Zariski decomposition $$P_t-uC=Q_{t,u}+N_{t,u},\qquad 0\le u\le\tau_t,$$ where $Q_{t,u}$ is nef and $N_{t,u}$ is effective with negative definite intersection matrix on its nonempty support. The number $\tau_t$ is the pseudoeffective threshold. Its coefficients satisfy $Q_{t,u}\cdot N_{t,u}=0$. Assume that $C$ is not a component of $N_{t,u}$ in the interior of the range and that the restricted moving series has, asymptotically, the complete series of degree $m\,Q_{t,u}\cdot C$. Set $$d(t,u)=Q_{t,u}\cdot C,\qquad F_{t,u}=(J_t-c_tC+N_{t,u})|_C.$$ The fixed divisor in this last expression is an actual divisor on $C$, not its degree. Under the section and convergence hypotheses above, one obtains (4.3)$$\begin{aligned} S(W;C)&=\frac3V\int\left(c_tP_t^2+ \int_0^{\tau_t}Q_{t,u}^2\,du\right)dt, \label{az:eq:surface-curve}\end{aligned}$$

(4.4)$$\begin{aligned} S(W^C;q)&=\frac3V\iint d(t,u)^2\,du\,dt +\frac6V\iint d(t,u)\mathop{\mathrm{ord}}_qF_{t,u}\,du\,dt. \label{az:eq:surface-point} \end{aligned}$$ For the first identity sum the dimensions of the filtered surface spaces, or integrate their volumes. For the second, the curve piece has $md(t,u)+O(1)$ sections and mean order $m\mathop{\mathrm{ord}}_qF_{t,u}+md(t,u)/2+O(1)$ by Lemma 3.1. Multiplying and summing proves the formula. There are two filtration indices, so order $m^2$ curve pieces; the numerator has order $m^4$, as does $mN_m$.

If only containment in these complete curve spaces is known, their weight sums give upper bounds with the same normalization. To claim the equalities, it is necessary to prove the stated section identification, an appropriate almost-completeness statement, or an $o(m^3)$ total dimension deficit. Checking the intersection numbers alone does not establish any of them.

To apply surface adjunction we require $(T,\Delta+C)$ to be plt near the centre, as in Proposition 2.7. Its estimate divides the two moments by different discrepancies. When $C$ is absent from $\Delta$, the first numerator is $A_{T,\Delta}(C)=1$. At $q$, the numerator is $1-\operatorname{coeff}_q\Delta_C$, where $(K_T+\Delta+C)|_C=K_C+\Delta_C$. If $\Delta=dB$, with $B$ smooth and transverse to $C$ at $q$, this is $1-d$, not one. If the surface is a quotient, the different includes the quotient contribution as well. The boundary and the fixed divisor $F_{t,u}$ have different roles and neither is absorbed into the other.

4.3 Total transforms when the flag is defined downstairs

Suppose the series was pulled back by $r:T\to T_0$ and the curve used for refinement is a rational Cartier curve $B\subset T_0$. Write $r^*B=C+Z$, where $C$ is its strict transform. The first moment is still (4.3), since $\mathop{\mathrm{ord}}_B=\mathop{\mathrm{ord}}_C$. The fixed divisor for the downstairs restriction, however, is (4.5)$$\label{az:eq:total-transform-correction} F_{t,u}^{\mathrm{down}}= \bigl(J_t-c_tC+N_{t,u}-(c_t+u)Z\bigr)|_C.$$ To see this, divide a section of order $m(c_t+u)$ along $B$ by the same power of a local equation of $B$, before restricting. On $T$ that equation has divisor $C+Z$. Subtracting only $C$ would leave the extra factor $m(c_t+u)Z$. Formula (4.5) removes precisely that factor. It is to be used in the geometric identification of the restricted rational sections, including their fractional factors.

For example, if a smooth point $p\in B$ is blown up, then $r^*B=C+E$ and $C\cap E$ is one point. Omitting the correction would increase the refined point moment there by $$\frac6V\iint d(t,u)(c_t+u)\,du\,dt=S(W;B).$$ The equality follows either from the dimension sum for the $B$-orders, or by integration of $-\partial_u Q_{t,u}^2=2d(t,u)$ on the chambers. This is the correction used in the infinitely near configurations of Family 1.9 and in Example ??. It is substantial even when the numerical class of the moving part is unchanged.

4.4 Blowing up a point of the surface

Let $p$ be a smooth point of a smooth surface $S$, let $D$ be an ample divisor, and let $\sigma:T\to S$ blow up $p$, with exceptional curve $E$. Write $\sigma^*D-uE=P_u+N_u$. The preceding calculation with just one filtration parameter gives (4.6)$$\label{az:eq:surface-point-blowup} S_D(E)=\frac1{D^2}\int P_u^2\,du,\qquad S(W^E;q)=\frac1{D^2}\int(P_u\cdot E)^2\,du +\frac2{D^2}\int(P_u\cdot E)\mathop{\mathrm{ord}}_q(N_u|_E)\,du.$$ For zero boundary, $A_S(E)=2$ and the different on $E$ is zero. Consequently $$\delta_p(S;D)\ge \min\left\{\frac2{S_D(E)},\inf_{q\in E}\frac1{S(W^E;q)}\right\}.$$ The infimum must include every point of $E$, including intersections with all negative curves occurring in $N_u$. Computing only the first ratio proves no lower bound for $\delta_p(S;D)$. If a boundary $\Delta$ is present, the first numerator becomes $2-\operatorname{mult}_p\Delta$. Provided $(T,E+\sigma_*^{-1}\Delta)$ is plt, the boundary for the second step is $(\sigma_*^{-1}\Delta)|_E$, with the corresponding point numerators. If this pair is not plt, a different extraction or a further calculation is needed.

An alternative is to refine the full family $W$ by this exceptional curve immediately, using (4.3) and (4.4). This often improves taking the worst polarized-surface bound. In either approach this operation is a blow-up of a surface; it does not require a Zariski decomposition on the threefold blown up at $p$.