Section 9

The E6 point

Part II · Family 2.5: cubic-surface fibrationsPDF page 44

We retain the global section ring and the two filtration identifications from Section 8, now with the $E_6$ weights. In particular, Lemmas 8.9 and 8.10 already give $S_X(G_E)=177/16$ and $S(W;B)=11/16$ .

9.1 The exceptional surface and its tangent curve

Lemma 9.1. The weighted blow-up is of plt type, $A_X(G_E)=13$ , and its coarse exceptional surface is $G_E=\mathbb P(1,1,2)$ with coordinates $U=x$ , $V=z^2$ , $W=y^3$ . For $H_G=\mathcal O_{G_E}(1)$ one has $-G_E|_{G_E}=\tfrac16H_G,\quad \Delta_G=\tfrac12(V=0)+\tfrac23(W=0),\quad B=(U^2+UV+W=0)\sim2H_G.$ Moreover, $\alpha(G_E,\Delta_G;H_G)=1/2$ .

Proof. Lemma 8.12 applies with weights $(6,4,3)$ and gives discrepancy $6+4+3=13$ . A monomial of weight divisible by six has exponent of $y$ divisible by three and exponent of $z$ divisible by two. Thus the sixth Veronese is $\mathbb C[x,z^2,y^3]$ with regraded weights $(1,1,2)$ ; it gives the coarse surface and the normal class $H_G/6$ .

The generic quotient indices are two along $z=0$ and three along $y=0$ , and one along $x=0$ . They give the stated different. The equation of $B$ follows from the initial relation. As a check, $-(K_{G_E}+\Delta_G)=13H_G/6=13(-G_E|_{G_E})$ . Finally, Lemma 3.2 gives $\min\{1,1/2,2/3\}=1/2$ . ◻

Lemma 9.2. The curve $B$ is a smooth rational curve avoiding the singular vertex of $G_E$ . It is parametrized by $[U:V]$ , with $W=-U(U+V)$ . Let $Q_U,Q_{U+V},Q_V$ denote its points $U=0$ , $U+V=0$ , $V=0$ . Then $(G_E,\Delta_G+B)$ is plt along $B$ and $\Delta_B=\tfrac23Q_U+\tfrac23Q_{U+V}+\tfrac12Q_V.$

Proof. At the vertex $[0:0:1]$ , the equation of $B$ is nonzero. Away from that point it solves for $W$ and gives the claimed isomorphism with $\mathbb P^1$ . The three marked points are distinct. The boundary $(W=0)$ cuts out $Q_U+Q_{U+V}$ transversely, and $(V=0)$ cuts out $Q_V$ transversely. Thus near $B$ the pair is a smooth surface with a smooth coefficient-one curve and transverse boundary curves of coefficients less than one. It is plt, and ordinary adjunction gives the different. ◻

Lemma 9.3 (Residual degree in type $E_6$ ). The residual-degree average is $R_E=15/32$ .

Proof. The class of a section of $W_{m,j}$ is $jH_G/6$ , and removal of $B^q$ subtracts $2qH_G$ . Therefore $R_E=\tfrac16S_X(G_E)-2S(W;B) =\frac{177}{96}-2\cdot\frac{11}{16}=\frac{15}{32}.$ ◻

9.2 The exact curve series and its averages

Lemma 9.4 (Finite-degree series in type $E_6$ ). Fix $m,q$ and set $M=\min\{m,2m-q\}$ . Every standard monomial is uniquely written $y^\epsilon z^\eta U^iV^j b^q,\qquad 0\le\epsilon\le2,\quad0\le\eta\le1,\quad i+j=n.$ Its native weight is $6n+4\epsilon+3\eta+12q$ . Put $T=M-\epsilon-\eta$ . After removing $b^q$ and the residue factor $y^\epsilon z^\eta$ , its weight summand restricts to $\label{f25:eq:E-series} \langle U^{n-j}V^j:0\le j\le\ell\rangle =U^{n-\ell}H^0(\mathbb P^1,\mathcal O(\ell)),\qquad \ell=\min\{n,T-n\},\quad0\le n\le T.$ It is empty if $T<0$ . For a nonempty summand the dimension is $\ell+1$ , and its mean point orders, with the residue factor restored, are $\begin{array}{c|cccc} Q&Q_U&Q_{U+V}&Q_V&\text{other points}\\\hline \text{mean}&n-\ell/2+\epsilon/3&\ell/2+\epsilon/3&\ell/2+\eta/2&\ell/2. \end{array}$ The residual divisor class on $G_E$ is $(n+2\epsilon/3+\eta/2)H_G$ .

Proof. The standard-monomial condition is that the exponent of $y$ is at most two; splitting the exponent of $z$ into its parity and its even part gives the unique expression. The six residues $4\epsilon+3\eta$ are distinct modulo six. Thus $\epsilon,\eta$ are fixed in each summand of given native weight and given $q$ .

The ordinary-degree inequality from Lemma 8.4 is $i+2j\le M-\epsilon-\eta=T$ . Since $i+j=n$ , it is equivalent to $0\le j\le\min\{n,T-n\}$ . This proves (9.1). Its fixed part is $(n-\ell)Q_U$ , and the remaining series is complete. Lemma 3.1 gives the means before restoring the residue factor. Using Lemma 2.14 and $y^3=W=-U(U+V)$ , $z^2=V$ , that factor contributes $\epsilon/3$ at $Q_U,Q_{U+V}$ and $\eta/2$ at $Q_V$ . Its class is $(2\epsilon/3+\eta/2)H_G$ , proving the last assertion. The fixed degree plus the moving degree is $(n-\ell)+2\epsilon/3+\eta/2+\ell=n+2\epsilon/3+\eta/2$ , as required by $\deg(H_G|_B)=1$ . ◻

Example 9.5. Take $m=5$ , $q=0$ , $\epsilon=\eta=0$ , $n=3$ . Then $T=5$ and $\ell=2$ . The space is $\langle U^3,U^2V,UV^2\rangle=U H^0(\mathbb P^1,\mathcal O(2))$ . Its three orders at $Q_U$ are $1,2,3$ , with mean two. At any other point an adapted basis has orders $0,1,2$ , with mean one. Thus the extra vanishing at $Q_U$ comes from the explicit fixed factor, rather than from a separate decomposition on the threefold.

Lemma 9.6 (Positivity in type $E_6$ ). The two refined series have bounded support and contain ample series, so both inequalities of Proposition 2.7 apply.

Proof. Use the native weight $J$ and put $M=\min\{m,2m-q\}$ . Here the second support cone is $\label{f25:eq:E-support} \mathcal C_E=\{(m,J,q):m\ge0,\ 0\le q\le2m,\ 12q\le J\le12q+6M\}.$ Containment follows from the standard monomials. To prove equality and eventual nonvanishing, take an integral interior point $(m_0,J_0,q_0)$ and write $M_0=\min\{m_0,2m_0-q_0\}$ . For each $k$ , choose the unique pair $0\le\epsilon_k\le2$ , $0\le\eta_k\le1$ such that $4\epsilon_k+3\eta_k\equiv kJ_0\pmod6$ . Set $n_k=\frac{kJ_0-12kq_0-4\epsilon_k-3\eta_k}{6},\qquad T_k=kM_0-\epsilon_k-\eta_k.$ The strict inequalities $0<J_0-12q_0<6M_0$ imply that $n_k$ and $T_k-n_k$ both grow linearly. Hence the interval in Lemma 9.4 is nonempty for every sufficiently large $k$ . Density of rational interior rays proves (9.2).

The first support cone is its projection $0\le J\le24m$ . For any interior ray in this projection, choose $q_0$ just below $J_0/12$ to lift it to an interior ray of (9.2), and approximate $kq_0$ by integers. The bounded rounding and residue errors preserve eventual nonvanishing, just as above. Thus both series have bounded support with nonempty interior and are eventually nonzero on every integral interior ray.

For an ample subseries on $G_E$ , take $(m,J)=(12k,12k)$ . Every weighted polynomial of weight $12k$ has ordinary degree at most $4k\le m$ and is a global initial form. These constitute $H^0(G_E,\mathcal O_{G_E}(2k))$ , where $2H_G$ is ample Cartier. For the curve, take $(m,J,q)=(12k,24k,k)$ ; then $\epsilon=\eta=0$ , $n=\ell=2k$ , and the series is exactly $H^0(B,\mathcal O_B(2k))$ . Both rays are interior. Together with the plt assertions in Lemmas 9.1 and 9.2, this verifies all the hypotheses of Proposition 2.7. ◻

Lemma 9.7 (Limiting point orders in type $E_6$ ). One has $S(W^B;Q_U)=5/16$ and $S(W^B;Q)=5/32$ for every $Q\ne Q_U$ .

Proof. Rescale $q,n$ by $m$ and set $M(q)=\min\{1,2-q\}$ . The limiting dimension function is $\ell(n,q)=\min\{n,M(q)-n\}$ for $0\le n\le M(q)$ . The normalizing integral is $I_E=\int_0^2\int_0^{M(q)}\ell\,dn\,dq =\frac14\int_0^2M(q)^2dq=\frac13.$ There are six residue classes, so the total dimension is $6I_Em^3+O(m^2)=2m^3+O(m^2)$ . The common factor six cancels in normalized means. The limiting mean order is $n-\ell/2$ at $Q_U$ and $\ell/2$ at every other point. Splitting the inner integrals at $n=M/2$ gives $\int_0^M n\ell\,dn=\frac{M^3}{8},\qquad \int_0^M\ell\left(n-\frac\ell2\right)dn=\frac{M^3}{12},\qquad \int_0^M\frac{\ell^2}{2}\,dn=\frac{M^3}{24}.$ For example, the middle integral is $\int_0^{M/2}n^2/2\,dn+\int_{M/2}^M(M-n)(3n-M)/2\,dn=M^3/12$ . Multiplying the last two values by $\int_0^2M(q)^3dq=5/4$ and dividing by $I_E=1/3$ gives $5/16$ and $5/32$ . The first gives the residual average $15/32$ , agreeing with Lemma 9.3. Similarly, $S(W;B)=I_E^{-1}\frac14\int_0^2qM(q)^2dq=11/16$ .

All residue corrections in Lemma 9.4 are bounded, uniformly in $Q$ . Their total contribution is $O(m^3)$ by the same dimension estimate as in the proof of Lemma 8.20, so they disappear after division by $mN_m$ . ◻

9.3 The E6 estimate

Lemma 9.8. One has $\delta(G_E,\Delta_G;W)\ge16/15$ .

Proof. For centres not contained in $B$ , the residual-degree estimate and Lemmas 9.1 and 9.3 give $S(W;v)\le2(15/32)A_{G_E,\Delta_G}(v)=(15/16)A_{G_E,\Delta_G}(v)$ . For centres on $B$ , apply the second refinement using Lemma 9.6. The curve term is $16/11$ , and the point terms are $\begin{array}{c|c|c|c} Q&A_{B,\Delta_B}(Q)&S(W^B;Q)&A_{B,\Delta_B}(Q)/S(W^B;Q)\\\hline Q_U&1/3&5/16&16/15\\ Q_{U+V}&1/3&5/32&32/15\\ Q_V&1/2&5/32&16/5\\ \text{other points}&1&5/32&32/5. \end{array}$ The smallest bound is $16/15$ , which proves the assertion for every divisorial valuation on the surface. ◻

Proof of Theorem 7.3. The first refinement, Lemma 9.8, and the separate calculation of $S_X(G_E)$ give $\delta_P(X)\ge\min\left\{\frac{13}{177/16},\frac{16}{15}\right\} =\min\left\{\frac{208}{177},\frac{16}{15}\right\}=\frac{16}{15}.$ ◻

Remark 9.9. The two independent routes to the first expected vanishing agree: $2(95/96)+8(11/16)=359/48$ and $6(15/32)+12(11/16)=177/16$ . The left sides use the curve-series distributions; Lemma 8.9 uses only the weighted Hilbert function and the weight of the relation. This agreement checks the normalization of the native valuations against that of the coarse exceptional surfaces. Throughout both proofs, all new expected vanishing orders and fixed factors were obtained from actual graded section spaces, without a Zariski decomposition on the weighted threefold blow-ups.

Proof of Theorem 1.1 for Family 2.5. Suppose that $X$ is not K-stable and choose $F,P$ as in Lemma 7.1. Then $\delta_P(X)\le A_X(F)/S_X(F)\le1$ . This contradicts Theorem 7.2 or Theorem 7.3. ◻