Section 31

Comparison with a degree-two surface flag

Part V · Family 2.11: blow-ups of cubic threefolds along linesPDF page 122

The following calculation explains why the weighted point extraction above is needed. A small point term on an auxiliary surface does not suffice if the preceding curve term is too large.

Lemma 31.1 (The degree-two surface flag). Let a hyperplane section $V_0\subset V$ be smooth, meet $\ell$ transversally at $q$ , and contain no line through $q$ . Let $S\in|H|$ be its strict transform, let $e_q\subset S$ be the exceptional curve, and put $A=H|_S$ . Then $S$ is a smooth del Pezzo surface of degree two. The strict transform $C$ of the tangent plane cubic at $q$ is a $(-1)$ -curve of class $A-2e_q$ . For the flag $S\supset e_q\ni p$ , its constants are $S_X(S)=\frac38,\qquad S(W^S;e_q)=\frac{155}{144},\qquad S(W^{S,e_q};p)=\frac{127}{144}+\frac5{32}(Ce_q)_p.$

Proof. Blowing up a point on no line of a smooth cubic surface gives a degree-two del Pezzo surface. The tangent plane cubic is irreducible with a double point at $q$ , since a reducible tangent section would contain a line through $q$ . Its strict transform is the stated $(-1)$ -curve. Contracting it shows that $2A-3e_q=-K_S+C$ is nef (it is the pullback of the anticanonical class of the resulting smooth cubic surface).

For $0\le u\le1$ put $a=2-u$ . The first refined series is asymptotically complete of class $aA-e_q$ . To refine along $e_q$ , put $b=1+v$ . The positive parts are $\begin{array}{c|cc} b&P_{u,v}&N_{u,v}\\\hline {[1,3a/2]}&aA-be_q&0\\ {[3a/2,2a]}&(2a-b)(2A-3e_q)&(2b-3a)C. \end{array}$ Nefness in the first interval follows by expressing its classes as nonnegative combinations of $A$ and $2A-3e_q$ ; in the second the positive part is orthogonal to $C$ . The threshold is $b=2a$ , as tested by $2A-3e_q$ . Their squares are $3a^2-b^2$ and $3(2a-b)^2$ , while their degrees on $e_q$ are $b$ and $3(2a-b)$ . Consequently $\begin{split} S(W^S;e_q)&=\frac16\int_1^2\left(\frac72a^3-3a^2+\frac13\right)da =\frac{155}{144},\\ \frac16\int\!\int(P_{u,v}e_q)^2\,dv\,du &=\frac16\int_1^2\left(\frac32a^3-\frac13\right)da =\frac{127}{144},\\ \frac13\int\!\int(P_{u,v}e_q)\operatorname{coeff}_CN_{u,v}\,dv\,du &=\frac13\int_1^2\frac{a^3}{8}da=\frac5{32}. \end{split}$ This proves all three formulas. ◻

Remark 31.2 (The curve term cannot be discarded). Even when $p\notin C$ , the preceding flag gives only $\min\{8/3,144/155,144/127\}=144/155<1.$ Likewise, the refined threshold $24/29$ for the second-kind exceptional divisor is insufficient by itself. The weight- $(1,1,2)$ extraction in Lemma 29.7 resolves the remaining point case by computing a different refined series. No negative boundary is introduced to combine these unsuccessful flags: such a correction would require an adjunction statement that does not hold in general.