The exceptional ruling over an index-three point

Proof. For a positive integer $m$ divisible enough to make $-mK$ Cartier, let $\delta_{m,p}$ be the infimum of the local log canonical thresholds at $p$ of the $m$-basis divisors. Thus $$\delta_{m,p}(X)=\inf_{p\in c_X(F)}\frac{A_X(F)}{S_m(F)}.$$ One has $\delta_{m,p}\to\delta_p$. Indeed, [6, Corollary 2.10] gives $S_m(F)\le(1+\eta)S(F)$ for all valuations and all sufficiently large $m$, so $\delta_{m,p}\ge\delta_p/(1+\eta)$. Conversely, fix any divisor whose centre contains $p$ and use $S_m(F)\to S(F)$, then take the infimum over these divisors. This gives the opposite limiting inequality. Relative vanishing and base change identify the spaces of anticanonical sections with a free module over $\mathbb C[s]$. The equivariant structure determines a finite filtration of $H^0(X,-mK_X)$; its associated graded is $H^0(X_0,-mK_{X_0})$. Fix a prime divisor over $X$ whose centre contains $p$, and choose a basis compatible with both its valuation filtration and this finite filtration. Two finite filtrations always have a common compatible basis. Rescaling the resulting sections by powers of $s$ gives a family of bases whose central values form a basis of $H^0(X_0,-mK_{X_0})$.

The corresponding basis divisors specialize flatly. Lower semicontinuity of log canonical thresholds along the given section shows that the threshold of the original basis divisor at $p$ is at least the local $m$-basis threshold of $X_0$ at $p_0$. Since the original basis was adapted to the chosen valuation, taking the infimum over valuations gives the same inequality for the local $m$-basis thresholds. Passing to the limit as $m$ tends to infinity proves the assertion. ◻

Proof. Points on the distinguished surface $E$ already have this bound. Choose coordinates in which $$f=x^3y+\alpha x^4,\qquad g=y^4+g_1xy^3+g_2x^2y^2+g_3x^3y+g_4x^4,$$ and the centre has equation $uf+vg=0$. Consider the family $$u(x^3y+s\alpha x^4)+v\left(y^4+ \sum_{i=1}^4s^ig_ix^iy^{4-i}\right)=0.$$ For $s\ne0$ the substitutions $x'=sx$, $y'=y$, $u'=u$, $v'=s^3v$ identify its member with the original curve. The family of blow-ups is equivalently the family of divisors of class $h+2H$ in $\mathbb F_1\times\mathbb P^2$ obtained from the net of quadrics. Here $h$ is the pullback of a line from $\mathbb P^2$ and $f_1$ is a ruling of $\mathbb F_1$. Consequently its relative anticanonical class is the restriction of $h+f_1+H$, which is ample.

The central net is $$\langle Y_0Y_2-Y_1^2,\ Y_1Y_2,\ Y_0^2\rangle.$$ Its incidence threefold has just one ordinary double point, over $[Y_0:Y_1:Y_2]=[0:0:1]$ and the coefficient of $Y_0^2$. Indeed, on the corresponding affine chart its quadratic part is $as+bt+s^2$, a nondegenerate quadratic form in four variables. The blow-up of the smooth fibre of the first generator takes place away from this point. Thus the central member is a nodal Fano threefold. The distinguished surface remains smooth.

We record the marked point carefully. On $u=Y_0=1$ put $x=Y_1$, $c=Y_2$, $z=c-x^2$, and $$a=\frac{xc+\alpha c^2+tG(x,c)}{z}.$$ The point $p_*$ has $a=0$. The given point has a finite value $a=a_0\ne0$, since the value infinity is the point on $E$. Under the displayed isomorphism the original exceptional coordinate and the coordinate on the member over $s$ satisfy $a'=s a$. Hence the marked point on that member has $a=a_0/s$ and specializes to the point of $E$ over $x=t=0$. In particular, it does not specialize to $p_*$.

The ruling of $E$ through this limit lies over $x=0$. The additional vertical component of the central fixed curve lies over $x=\infty$, so these two curves are disjoint. Lemma 120.2 therefore applies: its three expected orders are $17/24$, $11/12$ and at most $7/8$, and hence $\delta_{p_0}(X_0)\ge12/11$. Applying the preceding specialization lemma finishes the proof. ◻