Section 5

Identifying the section spaces and taking limits

Part I · Abban–Zhuang theoryPDF page 25

Lemma 5.1 (Identifying a geometric filtration). Let $P$ be a polynomial ring with an auxiliary grading whose homogeneous components are finite dimensional. Give its variables integer weights, let $f_1,\ldots,f_s$ be homogeneous for the auxiliary grading, and let $g_i$ be the lowest-weight form of $f_i$ . If $g_1,\ldots,g_s$ form a regular sequence, the associated graded ring for the induced quotient filtration is $P/(g_1,\ldots,g_s)$ .

Suppose that the assigned weights are the geometric orders of the generators. Assume that $P/(g_i)$ injects into its localization at homogeneous elements of weight zero that are units at the point. If the localized map to the geometric associated graded ring is injective, the quotient filtration equals the geometric filtration on each fixed auxiliary degree.

Proof. Filter the Koszul complex of the $f_i$ , shifting the filtration on its generators by the weights of the $g_i$ . Its associated graded complex is the Koszul complex of the $g_i$ , which is exact in positive homological degrees by regularity. In each auxiliary degree the filtration is finite, so exactness of the associated graded complex makes the filtered complex strict. Its zeroth associated graded quotient is therefore $P/(g_i)$ .

The assigned geometric orders give a natural map from this ring to the geometric associated graded ring. It becomes injective after the stated localization. Since the source has no torsion with respect to the elements inverted, it is already injective before localization. Thus a nonzero initial form cannot disappear on the local chart, and no additional cancellation can increase the order of a section. ◻

Lemma 5.2 (Weighted quotient geometry). For primitive positive weights $(w_1,w_2,w_3)$ at a smooth point, the weighted extraction is of plt type and $A_X(G)=w_1+w_2+w_3$ . The coefficient of the different along $x_k=0$ is $1-1/\gcd(w_i,w_j)$ , where $\{i,j,k\}=\{1,2,3\}$ .

Proof. On the $i$ th chart use the cover $x_i=s^{w_i}$ , $x_j=s^{w_j}y_j$ , $x_k=s^{w_k}y_k$ , with cyclic action of weights $(1,-w_j,-w_k)$ . Primitivity rules out divisorial ramification on the threefold cover. Thus the smooth plt pair with boundary $s=0$ descends. The Jacobian has order $w_1+w_2+w_3-1$ , giving the discrepancy. Restriction to the exceptional surface has generic ramification index $\gcd(w_i,w_j)$ along the indicated coordinate curve, giving the different. Relative ampleness follows from the weighted Proj construction. ◻

Lemma 5.3 (A monomial degeneration bound). Let $(T,\Delta)$ be a projective toric surface with torus-invariant boundary, and let $W$ be a multigraded series with bounded support. Suppose that, under a one-parameter subgroup of its torus, each finite-dimensional summand has a monomial initial space. Write the normalized sum of the zero divisors of all these initial monomials as $D_m^0=\sum_i c_{i,m}D_i,\qquad c_i=\lim_{m\to\infty}c_{i,m},$ where $D_i$ are the invariant prime divisors. The normalization is $mN_m$ , as before. Then $\delta(T,\Delta;W)\ge \min_{c_i>0}\frac{1-\operatorname{coeff}_{D_i}\Delta}{c_i}.$ The same conclusion holds for a succession of torus degenerations.

Proof. Fix $m$ and a divisorial valuation $v$ . First perturb the one-parameter subgroup. Only finitely many characters occur in the relevant spaces, so choose an integral torus weight $\rho$ distinguishing them and replace the original weight $\lambda$ by $N\lambda+\rho$ , with $N$ sufficiently large. The $\lambda$ -initial spaces are already monomial, so this refinement does not change them, and their monomials now have distinct weights. Now choose a basis of each summand compatible with $v$ and with this perturbed weight filtration, using [4, Lemma 3.1]. Its initial terms are distinct and form precisely the monomial basis of the initial space. The perturbation must precede the choice of basis: a basis compatible with the old weights need not be compatible with the new ones.

The resulting basis divisor degenerates to $D_m^0$ . A toric pair is log canonical if its coefficients along all invariant divisors are at most one: the discrepancies of further toric divisors are nonnegative linear combinations of the corresponding discrepancies, and a toric resolution checks log canonicity. Hence $(T,\Delta+\lambda_mD_m^0)$ is log canonical for $\lambda_m=\min_{c_{i,m}>0}(1-\operatorname{coeff}_{D_i}\Delta)/c_{i,m}$ . All nonzero fibres of the degeneration are isomorphic as pairs, since the torus preserves $\Delta$ . Lower semicontinuity of log canonical thresholds therefore gives the same bound for the original basis divisor. Its basis is compatible with $v$ , so $A_{T,\Delta}(v)\ge\lambda_m S_m(W;v)$ . The bound is independent of $v$ . Passing to the limit and taking the infimum over $v$ proves the assertion. For successive degenerations combine the successive weights with sufficiently dominant integral coefficients at the fixed finite level, and then choose the compatible bases.

For rational divisor classes on the coarse weighted planes, use the effective rational zero divisors as in Proposition 2.7. Their bounded fractional coordinate factors are retained in $D_m^0$ ; they contribute $O(1/m)$ to its normalized coefficients. Clearing Cartier indices does not remove the other residue classes or change $N_m$ . ◻

5.1 Hilbert series with a weight variable

Once an assigned grading has been identified with the geometric filtration, its Hilbert series computes the expected order directly. If $h_{m,j}$ is the dimension in degree $m$ and weight $j$ , set $H(z,q)=\sum h_{m,j}z^mq^j$ . Coefficientwise differentiation gives $N_m=[z^m]H(z,1),\qquad \sum_j jh_{m,j}=[z^m] \left.q\frac{\partial H}{\partial q}\right|_{q=1}.$ Dividing the second quantity by $mN_m$ and taking its limit gives $S$ , by the adapted-basis formula. For a polynomial ring with generator degrees and weights $(d_i,w_i)$ , divided by a homogeneous regular sequence with degrees and weights $(e_k,r_k)$ , the series is $H(z,q)=\frac{\prod_k(1-z^{e_k}q^{r_k})} {\prod_i(1-z^{d_i}q^{w_i})}.$ This follows from the weighted Koszul resolution. A multigrading is handled by replacing $z^{d_i}$ by the corresponding monomial and then extracting the required diagonal coefficients. The weights of the equations contribute to the numerator and cannot be omitted.

For example, the diagonal ring of three equations of bidegree $(1,1)$ in $\mathbb P^3\times\mathbb P^3$ , used in Family 2.12, has $N_m=(2m+1)(5m^2+5m+3)/3$ . If the eight coordinate weights sum to $A$ and the three equation weights sum to $B$ , logarithmic differentiation and extraction of the diagonal give leading weight coefficient $35A/24-5B/6$ . Therefore its limiting mean is $7A/16-B/4$ . Lemma 33.5 records the full polynomial calculation. It computes a geometric expected order only after the identification of the actual initial ring; an arbitrary weighted degeneration of the equations would not do so.

5.2 Bounding an ordinary point order by a flag

For a flag $p\in C\subset S\subset X$ smooth at $p$ , choose regular local parameters $x,y,z$ with $S=(x=0)$ and $C=(x=y=0)$ . For a section $s$ in a local trivialization, let $i$ be its order along $S$ ; let $j$ be the order along $C$ of $(s/x^i)|_S$ ; and let $k$ be the order at $p$ of the remaining nonzero section on $C$ after its $y^j$ factor is removed. These three nonnegative integers are successive, defined vanishing orders. The formal expansion of $s$ contains the term $x^iy^jz^k$ with nonzero coefficient, so $\operatorname{mult}_p(s)\le i+j+k$ .

On the finite-dimensional space $H^0(X,mL)$ , lexicographic order on these triples defines one filtration, with finitely many steps. Choose a common basis for it and ordinary multiplicity at $p$ . Let $a_m,b_m,c_m$ be the sums of its three respective orders, each divided by $mN_m$ . If their limits are $a,b,c$ , then the ordinary exceptional divisor of $\operatorname{Bl}_pX$ satisfies $S_L(E)\le a+b+c$ . There are exactly two filtrations in this argument; one is the entire successive flag filtration. It is not an assertion about simultaneous splitting of three unrelated filtrations. Lemma 14.3 applies this observation to the cubic-surface flags.

5.3 Containing spaces and the dimensions of the actual spaces

Lemma 5.4 (Keeping the missing dimensions). Let $U\subset V$ have the induced filtration, and suppose all weights of $V$ lie in $[a,b]$ . If $w$ denotes the sum of weights of an adapted basis and $d=\dim V-\dim U$ , then $w(V)-bd\le w(U)\le w(V)-ad.$ The same statement holds for a direct sum of pieces, with the corresponding $a,b,d$ in each piece.

Proof. The quotient filtration on $V/U$ has $d$ weights in $[a,b]$ . Additivity of the weight sums gives $w(V)-w(U)=w(V/U)\in[ad,bd]$ . ◻

If the containing space is larger by $o(m^3)$ and its weights are $O(m)$ , the discrepancy in normalized moments tends to zero. If its extra dimension has order $m^3$ , it can change the leading moment. The last lemma gives a correction even if the smaller space is not spanned by monomials: the induced filtration, or an invariant subspace of a diagonal action, suffices. Its dimension is always the dimension of the actual flat limit, not the dimension of the complete linear system on the limiting scheme.

There are two different ways in which we certify that a calculation uses the whole space. For equality in every degree we construct a map from a candidate initial algebra onto the relevant initial algebra, or a space of genuine initial sections inside it, and compare their Hilbert functions. An equality of dimensions then proves equality of spaces. For asymptotic moments it suffices that the total dimension deficit is $o(m^3)$ , provided all weights are linearly bounded. The latter statement does not assert an equality of graded algebras or surjectivity of each restriction map. These two arguments occur respectively in Lemma 94.2 and Lemma ??.

5.4 Certifying all moments from actual leading terms

The following algebraic observation is useful when cancellations produce additional leading terms beyond those of the chosen generators. A valuation with one-dimensional leaves admits an adapted basis of each finite-dimensional space whose value vectors are pairwise distinct. Here “one-dimensional leaves” means that each nonzero associated graded piece for this valuation has dimension one. Orders are additive under multiplication.

Lemma 5.5. Let $A=\bigoplus_{m\ge0}A_m$ be a domain and suppose $\dim A_m=cm^n+O(m^{n-1})$ . A valuation with one-dimensional leaves takes its values on $A_m\setminus\{0\}$ in a rank- $n$ lattice $\Lambda$ , and all these values have norm at most $Cm$ . Suppose finitely many actual sections have degree and value pairs $(m_i,v_i)$ , with $m_i>0$ , whose generated group is $\mathbb Z\times\Lambda$ . Set $P=\operatorname{conv}\{v_i/m_i\}$ . Give a fundamental parallelepiped of $\Lambda$ volume one. If the volume of $P$ in this measure is $c$ , then $\lim_{m\to\infty}\frac1{m\dim A_m} \sum_{v\in\nu(A_m\setminus\{0\})}v =\frac1c\int_P x\,dx.$

Proof. Let $\Gamma$ be the semigroup generated by the displayed degree and value pairs. Its saturation is the lattice points of its rational cone. This saturation is finite as a $\Gamma$ -module. Choose finitely many module representatives, write each as a difference of two elements of $\Gamma$ , and add the negative parts. The resulting $\eta\in\Gamma$ satisfies $\eta+\overline\Gamma\subset\Gamma$ . Therefore the lattice points missing from a degree- $m$ slice lie in bounded-width boundary layers. Their number is $O(m^{n-1})$ . Lattice-point counting gives $\#\Gamma_m=cm^n+O(m^{n-1})$ and $\sum_{v\in\Gamma_m}v=m^{n+1}\int_Px\,dx+O(m^n)$ . Products of the certified sections show that these vectors all occur in $A_m$ . The full space has only $O(m^{n-1})$ more values; each has norm $O(m)$ . Their contribution to the sum is $O(m^n)$ . Division by $m\dim A_m$ proves the claim. ◻

The group condition includes the degree coordinate; full real span is insufficient. An index in the value lattice changes the volume. Negative value coordinates cause no problem if their absolute values are bounded by $Cm$ . We also sometimes use an increasing sequence of polytopes from certified sections, each satisfying the same full-group condition, whose volumes tend to $c$ . Fix one polytope first, bound the contribution of all remaining values by $C$ times its missing volume, let $m$ tend to infinity, and only then enlarge the polytope. This gives the same conclusion. These are the precise dimension arguments used for the ordinary and weighted point extractions in Families 2.12 and 2.14; see Lemma 34.2. A list of leading monomials without this check need not determine an expected order.

5.5 Degenerating the series while preserving the boundary

The monomial lemma above uses an action preserving the boundary. More generally, let a one-parameter group act on a projective pair $(T,\Delta)$ and preserve $\Delta$ . Suppose its action linearizes the line bundles of the finite-dimensional pieces of a series. Fix a valuation $v$ and a degree $m$ . In each piece choose a basis compatible with the $v$ -filtration and with the filtration of this action. After rescaling the basis vectors by their weights, their limits are independent and form a basis of the Grassmannian limit of the piece. Thus the total basis divisor $D_m$ specializes to a basis divisor $D_m^0$ of spaces of exactly the same dimensions. For rational section classes, use a common Cartier multiple in forming these divisors and retain all fractional fixed factors.

If $(T,\Delta+\lambda_mD_m^0)$ is log canonical, lower semicontinuity of log canonical thresholds gives log canonicity for the nonzero fibres. These fibres are isomorphic as pairs because the action preserves $\Delta$ . Consequently $\lambda_m S_m(W;v)=\lambda_m v(D_m)\le A_{T,\Delta}(v).$ A bound for the limiting divisors independent of $v$ , followed by the limit in $m$ , bounds the stability threshold. This argument does not require the limiting spaces to be complete or monomial. Repeated weights allow non-monomial eigenvectors. A containing monomial space can be used to estimate their orders, but its extra dimensions must be treated as in Lemma 5.4. Moving the sections by an action which does not preserve the boundary instead degenerates a different pair and does not prove the asserted bound for $(T,\Delta)$ .

A frequent exceptional pair in this manuscript is $(\mathbb P^1\times\mathbb P^1,D/2)$ , where $D$ is a smooth curve of bidegree $(2,1)$ . Coordinates put it in the form $u=s^2$ on an affine chart, and the action $(s,u)\mapsto(\lambda s,\lambda^2u)$ preserves it. Its invariant prime curves are the four coordinate fibres and the closures of $u=c s^2$ , $c\ne0$ . The latter curves meet at the two ramification points of $D$ over the second factor. Away from those points the invariant boundary has simple normal crossings.

Lemma 5.6. At the origin of the smooth affine plane, an effective divisor $B=a(s=0)+b(u=0)+\sum_{i=1}^r c_i(u=\gamma_i s^2)$ with distinct nonzero $\gamma_i$ gives a log canonical pair if and only if $a\le1,\quad b\le1,\quad c_i\le1, \qquad a+2b+2\sum_i c_i\le3.$

Proof. Blow up the origin, then blow up the point where the strict transforms of $u=0$ and the parabolas meet the first exceptional curve. The resulting boundary has simple normal crossings. The two exceptional valuations have weights $(1,1)$ and $(1,2)$ on $(s,u)$ and discrepancies two and three. Their inequalities are $a+b+\sum c_i\le2$ and $a+2b+2\sum c_i\le3$ . The first follows from the second and $a\le1$ , by taking their average. Together with the prime coefficients at most one these are all inequalities on this resolution, proving both directions. ◻

For $D/2$ , the weighted valuation $(1,2)$ therefore has discrepancy $3-\tfrac12\cdot2=2$ . The first ordinary blow-up alone does not separate the curves tangent to $D$ , which is why the second blow-up or the corresponding weight- $(1,2)$ extraction is useful. This elementary resolution explains the boundary calculation used in Families 2.9, 2.13, 2.16 and 3.8.

Example 5.7 (Missing sections in Family 2.13). For a double conic in Family 2.13 the actual spaces on the exceptional product surface are ideal-power series for a length-two subscheme of $D$ ; their identification is proved in Lemma 41.4. A boundary-preserving specialization sends that scheme to $(u,s^2)$ , except for the separately treated pair of ramification points. In degree $m$ , a limiting piece embeds in the monomial space $\langle s^iu^l:0\le i\le m-\epsilon,\ 0\le l\le r, i+2l\ge2\max(0,r-m)\rangle, \qquad\epsilon\in\{0,1\}.$ Put $\rho=r/m$ . The monomial space has limiting polygon $0\le a\le1$ , $0\le b\le\rho$ , $a+2b\ge2\max(0,\rho-1)$ . Its total dimension, normalized by the actual $N_m$ , tends to $41/40$ , not one. Its normalized $(1,2)$ -weight sum is $163/80$ , which is larger than the discrepancy two just computed.

For $3/2<\rho\le2$ , the excess area over the actual dimension is $(\rho-3/2)^2$ , and every containing monomial has weight at least $2m(\rho-1)$ . Lemma 5.4 subtracts $\frac35\int_{3/2}^2 2(\rho-1)(\rho-3/2)^2\,d\rho=\frac7{160}.$ The actual weight sum is consequently at most $163/80-7/160=319/160<2$ . At the opposite ramification point it is at most $253/160$ . The four coordinate coefficients are at most $17/32,79/160,241/320,87/160$ , and every $(2,1)$ -curve coefficient is at most $81/320$ ; these integrals are displayed in Lemma 41.7. At $\lambda=320/319$ , the largest coordinate coefficient is at most $241/319<1$ and the coefficient on $D$ is at most $1/2+81/319<1$ . The weighted inequalities of Lemma 5.6 hold as well. Using slightly smaller $\lambda$ at finite degree and then passing to the limit gives $\delta(F,D/2;W)\ge320/319$ in this case. The pair of ramification points gives the stronger bound $8/7$ . Thus the strict estimate uses both the different and the number of sections missing from the containing space.