On local stability threshold of del Pezzo surfaces

We complete the classiﬁcation of local stability thresholds for smooth del Pezzo surfaces of degree 2. In particular, we show that this number is irrational if and only if the tangent plane at the point intersecting the surface is the union of a line and a smooth cubic curve meeting transversally at the point.

In this paper, we will focus on the study δ, mostly in the local situation.It was originally conjectured that δ(X) is rational for K-polystable Fano varieties, and was proven by Liu, Xu, and Zhuang in [LXZ22] when δ(X) < (n + 1)/n where n is the dimension of X.However, the local stability threshold, δ p (X), is more mysterious.Its rationality can be unpredictable.Note that δ(X) is the infimum of δ p (X) for all p ∈ X.The only known example where the local delta invariant is irrational is in cubic surfaces (see [AZ22,Lemma A.6]).The rationality of δ p (X) is well-established for all points p in the case of del Pezzo and weak del Pezzo surfaces with a degree equal to or greater than 4 (refer to [ACC + 23, §2], [Aka23,Den23]).It is expected that irrational δ p exists only for certain points p in del Pezzo surfaces of degrees 1, 2, and 3.In this paper, we settle this in degree 2 with the following theorem.
Theorem 1.1.Let X ⊂ P(1, 1, 1, 2) be a smooth del Pezzo surface of degree 2 and let p 0 = (x, y, z, w) ∈ X be a closed point with w ̸ = 0. Assume that there is a unique (-1)-curve L passing through the point p 0 .Then δ p 0 (X) = 6 71 (11 + 8 √ 3) and it is computed by taking the limit of a sequence of weighted blow ups at p 0 with wt(u) = a m and wt(v) = b m such that a m /b m → 2/ √ 3 when m → ∞, where u and v are the local coordinates such that L = {u = 0}.
Sketch of the proof.In order to prove that the local stability threshold at p 0 (as in Theorem 1.1) is irrational, we follow these steps: First, we take E a,b to be the exceptional divisor of a certain weighted blowup at p 0 with weights (a, b).The choice of (a, b) becomes clear in §5.Using that exceptional divisor we compute an upper bound for δ p 0 (X).The difficulty here is to compute the expected vanishing order of the anticanonical divisor of X with respect to E a,b , which requires a Zariski Decomposition and a careful choice of negative curves in the weighted blowup of X.This is the main bulk of the work in section §4.2.Then, we use techniques from [AZ22] to find lower bounds for δ p 0 (X), which requires a where the sum runs over f -exceptional divisors E. Recall that a(E; X, D) is a rational number independent of the morphism f and depends only on the discrete valuation which corresponds to E. This is called the discrepancy of the pair (X, D) at E. The log discrepancy of the pair (X, D) at E is defined to be A X,D (E) = a(E; X, D) + 1.
There is a more general definition of log discrepancy in [LXZ22, Def.2.5].
Since we are working with Fano surfaces, it is essential to recall the characterization for ampleness and other properties of divisors which you can find in different references such as [Mat02,Laz04].Definition 2.3.Let r > 0 and a 1 , ..., a n be integers and let x 1 , ..., x n be coordinates on A n .Suppose that µ r acts on A n via: where ε is a fixed primitive r-th root of unity.A singurality Q ∈ X is a quotient singularity of type 1 r (a 1 , ..., a n ) if (X, Q) is isomorphic to an analytic neighbourhood of (A n , 0)/µ r , where µ r is the set of all r-th roots of the unity.
Apart from the quotient singularities, we can also define the following ones: Definition 2.4.[Kol13, Def.2.8] Let (X, D) be a log pair where X is a normal variety of dimension ≥ 2 and D = a i D i is a sum of distinct prime divisors, where a i are rational non-negative numbers, all ≤ 1. Assume that m(K X + D) is Cartier for some m > 0. We say that (X, D) is Here klt is short for 'Kawamata log terminal', and plt for 'purely log terminal'.
Definition 2.5.We say that X is log Fano if there exists a divisor D such that −(K X + D) is ample and (X, D) is Kawamata log terminal.
Definition 2.6.Let (X, ∆) be a log pair and let F be a divisor over X, i.e.F is a divisor in a variety Y such that π : Y → X is a birational morphism.When F is a divisor on X we write ∆ = ∆ 1 + aF where F ⊈ Supp(∆ 1 ); otherwise let ∆ 1 = ∆.
(i) F is said to be primitive over X if there exists a projective birational morphism π : We call π : Y → X the associated prime blowup (it is uniquely determined by F ).
(ii) F is said to be of plt type if it is primitive over X and the pair (Y, ∆ Y + F ) is plt in a neighbourhood of F , where π : Y → X is the associated prime blowup and ∆ Y is the strict transform of ∆ 1 on Y .When (X, ∆) is klt and F is exceptional over X, π is called a plt blowup over X.

Bounds for the stability threshold
In this paper, we do not see the original definition of K-stability in terms of test configurations given in [Tia97,Don02].Instead, we focus on a simpler characterization of K-stability for Fano surfaces introduced in [FO18], which we use later in our computations.Recall that our setup is del Pezzo surfaces with at most klt singularities.However, these results can be generalised to higher dimensions.
Definition 2.7.Let X be a normal variety of dimension n such that K X is a Q-Cartier divisor.Let f : Y → X be a birational morphism and take E a prime divisor on Y .We define the expected vanishing order as follows: τ (E) is called the pseudoeffective threshold.Now, we can introduce the notions to characterize the K-stability for Fano varieties.
Definition 2.8.Let X be a Fano variety with at most klt singularities and let −K X be its anticanonical divisor.The (adjoint) stability threshold (or δ-invariant) of X is defined as where the infimum runs over all divisors E over X.
We say that a divisor E over X computes δ(X) if it achieves the infimum in (1).
There is also a local version of the stability threshold, which we focus on in this paper.
Definition 2.10.[AZ22] Let X be a Fano variety with at most klt singularities and let −K X be its anticanonical divisor.Let p be a closed point of X.We set where the infimum runs over all divisors E over X whose center contains p.
Similar notions can be defined for N × N r -graded linear series with bounded support on X.Details can be found in [AZ22, Section 3], but here, we just recall the results we use.
Let us fix a klt pair (X, ∆), some Cartier divisors L 1 , ..., L r on X and an N r -graded linear series V− → • associated to the L i 's such that V− → • contains an ample series and has bounded support.Let F be a primitive divisor over X with associated prime blowup π : Y → X. Assume that F is either Cartier on Y or plt type and let W− → • be the refinement of V− → • by F .Theorem 2.11.[AZ22,Theorem 3.3] Let F be a primitive divisor over X with associated prime blowup π : Y → X.Let Z ⊂ X be a subvariety, and let Z 0 be an irreducible component of Z ∩ C X (F ).Let ∆ Y be the strict transform of ∆ on Y (but remove the component F as in Definition 2.6) and let where the infimums run over all subvarieties Z ′ ⊆ Y such that π(Z ′ ) = Z 0 .
The following Theorem is a direct consequence of [AZ22, Theorem 3.2] and a simplification of [ACC + 23, Remark] in the surface case.It allows us to simplify the problem of finding a lower bound for the local stability threshold, using local stability thresholds of lower dimensional varieties.
Theorem 2.12.Let p be a point in a del Pezzo surface X with at most klt singularities, let π : Y → X be a plt blowup of the point p, and let E be the π-exceptional divisor.Then (Y, E) has purely log terminal singularities, so that there exists an effective divisor ∆ E defined by where τ (−K X ; E) denotes the pseudo-effective threshold as in Definition 2.7, let us denote by P (t) the positive part of the Zariski decomposition of the divisor π * (−K X ) − tE, and let us denote by N (t) its negative part.Let W E •,• be the refinement of −K X by E (see [AZ22,§2.4]for the definition).Then where for every x ∈ E we have This result gives a concrete recipe for what the flag of subvarieties of X is, which is a Mori Dream space [HK00], and we apply it in Theorem 5.2.Moreover, it is one of the key points for the proof of the Main Theorem 1.1.

Del Pezzo surfaces of degree 2
As we mentioned in the introduction, the objects of study in this paper are the del Pezzo surfaces of degree 2. These surfaces are contained in weighted projective spaces and are defined as follows: Definition 3.1.A del Pezzo surface or Fano surface is a 2-dimensional Fano variety, i.e. a non-singular projective algebraic surface with ample anticanonical divisor class, ω −1 X = −K X .The degree d of a del Pezzo surface X is given by the self-intersection of the anticanonical divisor, (−K X ) 2 = d.Remark 3.2.All of them but P 1 × P 1 can be represented as the blowup of P 2 at r points in general position.In these cases, d = 9 − r.
From now on, we focus on the case of del Pezzo surfaces of degree 2, which is represented by X.
X can be realized as the surface in weighted projective space P(1, 1, 1, 2) with homogeneous coordinates x, y, z, w, given by the equation where G 2 (x, y, z) and G 4 (x, y, z) are weighted homogeneous polynomials of degrees 2 and 4, respectively.When we consider X over C, we may assume G 2 (x, y, z) = 0. Therefore, X is given by the equation: Notice that there exists ρ : X → P 2 a double cover of P 2 ramified over a (smooth) quartic curve R [Dol12], which is a canonical model of a curve of genus 3.

Cases covered in [ACC + 23]
In the proof of [ACC + 23, Lemma 2.15], we can find a classification of the points in a smooth del Pezzo surface of degree 2, i.e. if a point p ∈ X, it satisfies one of the following: (1) If we do an ordinary blowup of p we get a smooth del Pezzo surface of degree 1; (2) ρ(p) ∈ R, and C p is an irreducible nodal curve; (3) ρ(p) ∈ R, and C p is an irreducible cuspidal curve; (4) ρ(p) ∈ R, and C p is a union of two (-1)-curves that meet transversally; (5) ρ(p) ∈ R, and C p is a union of two (-1)-curves that are tangent at p; (6) ρ(p) / ∈ R, and p is contained in exactly one (-1)-curve; (7) ρ(p) / ∈ R, and p is contained in two (-1)-curves; (8) ρ(p) / ∈ R, and p is contained in three (-1)-curves; (9) the point p is a generalized Eckardt point.
where C p is the unique curve in | − K X | that is singular at p.And in each case δ p is computed to be: Notice that case (6) is left, when p is contained in a unique (−1)-curve.However, the book [ACC + 23] gives a bound for it, so we know that 40 19 ≥ δ p (X) ≥ 60 31 .Therefore, we focus on computing δ p (X) for this case.It follows directly from equation (3), that if ρ(p) / ∈ R then the last coordinate of p is non-zero.

Weighted blowup of X
In this section, we introduce some essential technical results for proving Theorem 1.1.Let π a,b : X a,b → X be the (a, b)-weighted blowup of a del Pezzo surface of degree 2 at a point p 0 ∈ X.As in Theorem 1.1, we are assuming that there is a unique (-1)-curve L passing through the point p 0 .We write their equations in local coordinates u and v such as L = {u = 0}, and wt(u) = a and wt(v) = b after the weighted blowup.).However, currently we do not have enough negative curves in the picture.To find a convenient expression, we search for another known algebraic variety that is birational to the weighted blowup.The idea is to find a linear equivalence in terms of the strict transform of L in that space along with other negative curves, and bring it back to the weighted blowup X a,b .

Notation
As in Theorem 1.1, let p 0 = (x 0 , y 0 , z 0 , w 0 ) ∈ X be a closed point with w 0 ̸ = 0. Let X a,b be the (a, b)-weighted blowup of X at p 0 .Let L be the unique (-1)-curve passing through p 0 .Let E be the exceptional divisor of the weighted blowup.We assume Since {γ k } is a decreasing sequence of natural numbers, there exists a k 0 ∈ N where γ k 0 = 1.Furthermore, we can choose k 0 such that for any other k ′ ∈ N where Let us denote so n = n m=1 j 2m−1 and se n = n m=1 j 2m and for n = 0, so 0 = se 0 = 0.

The resolution of X a,b followed by contractions to a weak degree 1 del Pezzo
An (a, b)-weighted blowup of a smooth surface has at most two quotient singularities p and q.
, respectively.Similarly, we define the following: The resolution of these singularities is analogous to the Euclidean algorithm.The resolution of the singularity p is achieved for some i 0 ∈ N such that d i 0 = 1 and similarly for q, the resolution will be complete when for some j 0 ∈ N we get c j 0 = 1.Remark 4.1.Each d k can be represented as also holds for the coefficients µ k and λ k , and From now on, we assume neither a nor b are equal to 1.If this is not the case, it is enough to omit the singularity that corresponds to that weight.We present an algorithm to determine the resolution of X a,b .It is achieved by repeatedly blowing up the singularities.

Resolution of p.
Let σ 1 : Xa,b → X a,b be the weighted blowup of X a,b at the quotient singular point p with suitable (natural) weight.Let E (1) be the exceptional divisor and let L and Ê be the strict transforms of L and E respectively.Since p is a . Therefore, we get the following self-intersections: After this new blowup, if d 1 = 1, Xa,b will only have a singularity left and we continue with the resolution of q.On the other hand, if d 1 ̸ = 1, Xa,b has a singularity of type 1 d 1 (1, d 2 ) (as well as q) in the intersection of E (1) with Ê and we iterate the process by blowing it up.Denote this new blowup by σ 2 : Ẋa,b → Xa,b .Let E (2) the exceptional divisor of the σ 2 blowup, where (E (2) ) 2 = − d 1 d 2 .Let Ė(1) and Ė be the strict transforms of E (1) and Ê, respectively.We get the following self-intersections: Since gcd(a, b) = 1, by the Euclidean algorithm there exists an i 0 ∈ N such that d i 0 = 1.Therefore, the resolution of p is achieved after i 0 steps.Let us denote f p : Xa,b → X a,b the resolution of p defined as For each of these blowups with suitable (natural) weight of a quotient singular point, let us denote by E (k) the exceptional divisor of σ k .Let Ě(k) , Ě, Ľ, and Č be the strict transforms of our divisors at the i 0 -th blowup.Then, we have the following self-intersections: Proof.By induction, assume it is true for the k-th blowup, (E) 2 = − µ k ad k .Let us check that it holds for the (k + 1)-th blowup.
Therefore, we get (E) 2 = − µ k+1 ad k+1 .In particular, since The values of i 0 , d i 0 −1 , and m k can be specified using the notation of Subsection 4.1.We have two possibilities: Resolution of q.Let τ 1 : •• X a,b → Xa,b be theweighted blowup of Xa,b at the quotient singular point q with suitable (natural) weight.Let F (1) be the exceptional divisor and E be the strict transforms of Ľ, Ě(k) and Ě respectively.Since q is a 1 a (1, c 1 ) quotient singularity, (F (1) ) 2 = − a c 1 .Also, since q is in Ě, its strict transforms is not isomorphic to the pullback: 1) .Therefore: X a,b is smooth and it is the resolution.On the other hand, X a,b has a singularity of type 1 c 1 (1, c 2 ) in the intersection of F (1) and E, and we iterate the process by blowing it up, with the blow up denoted by τ 2 : the exceptional divisor of the τ 2 blowup, where (F (2)  E, respectively.We get the following self-intersections: Since gcd(a, b) = 1 by the Euclidean algorithm there exist j 0 ∈ N such that c j 0 = 1.Therefore, the resolution of q is achieved after j 0 steps.Let us denote f q : X a,b → Xa,b the resolution of q defined as For each of these blowups with suitable (natural) weight of a quotient singular point, let us denote by F (k) the exceptional divisor of τ k .Let F (k) , E (k) , E, and L be the strict transforms of our divisors after the (i 0 + j 0 )-th blowup.Then, we have the following self-intersections: Remark 4.3.Notice that the self-intersections of E (k) and L are the same as their images through f q .
Proof.First we see that in the (i . And then we see that α j 0 µ i 0 − β j 0 λ i 0 = 1.As before, by induction on k, assume it is true for the i 0 + k-th blowup.Then let us see that it also holds for the k + 1-th.
Where for the first equation we use that 1 . And in particular, since c j 0 = 1 in the last blowup we get (E) 2 = −(α j 0 µ i 0 − β j 0 λ i 0 ).Finally, we need to show that α j 0 µ i 0 − β j 0 λ i 0 = 1.We know by definition that α j 0 a + β j 0 b = bµ i 0 + aλ i 0 = 1.So, a = n(µ i 0 − β j 0 ), b = n(α j 0 − λ i 0 ).However, by the choice of µ i 0 , λ i 0 , β j 0 and α j 0 , we conclude n = 1.Then, since The values of j 0 , c j 0 −1 and n k can be specified using the notation in §4.1.We have two possibilities: In the following picture, we can see all the divisors involved in the resolution of the singularities p and q when k 0 = 2n 0 .Notice that the subscripts represent the selfintersections.
• If k 0 = 2n 0 + 1, In the following picture, we can see all the divisors involved in the resolution of the singularities p and q when k 0 = 2n 0 + 1.As before, the subscripts represent the selfintersections.
In total, we contract i 0 + j 0 = i + j + k 0 k=1 j k (-1)-curves in both cases.Let g : X a,b → X the composition of all the contractions.The picture at the end is: ), and We see now that this X is a weak del Pezzo surface of degree 1, and π 1,1 is the ordinary blowup at the point p 0 ∈ X, where Proposition 4.5.The surface X is a weak del Pezzo surface of degree 1.
Proof.With each blowup we do for the resolution, we get an exceptional divisor, as we saw above, and these divisors are added with a nonpositive coefficient to the equivalence class of the anticanonical.For instance, with the weighted blowup we get Notice that the coefficients of the strict transforms of the existing divisors do not change and do not change in the successive ordinary blowups either.The same thing happens when we contract exceptional curves, the coefficients of the remaining exceptional curves do not change in the process.Therefore, to prove that (−K X ) 2 = 1.We need to find the coefficient e of (1) .We got F (1) , with the first blowup of the singular point q ∼ 1 a (1, c 1 ), where c 1 = −b + a.Therefore, after the ordinary blowup at q, σ : Y → X a,b , (we are abusing notation here since this is not the same order of resolution we used above, but the coefficient of F (1) still is the same since it is independent of the order of the blowup), we get the following where E is the strict transform of E: Moreover, notice that e = −1.

So we have that, (−K
we will be back in the original degree 2 del Pezzo surface we started with.Therefore, we can say that π 1,1 : X → X is the ordinary blowup of p 0 ∈ X. Remark 4.6.In a weak del Pezzo surface of degree 1, we have a birational involution called the Bertini involution, ι.As in the proof of Lemma 2.15 in [ACC + 23], the linear system | − 2K X | gives a morphism X → P(1, 1, 2) with the following Stein factorization: where υ is a contraction of all (−2)-curves in the surface X (if any), and ω is a double cover branched over the union of a sextic curve in P(1, 1, 2) and the singular points of P(1, 1, 2).Notice that X is smooth if and only if −K X is ample, and in that case, υ is an isomorphism.Furthermore, if the divisor −K X is not ample, then the surface X contains at most four (-2)-curves.The double cover X → P(1, 1, 2) induces an involution ι ∈ Aut(X) known as the Bertini involution.For detailed equations of the Bertini involution, check [Moo43].
), and with this involution it is straightforward to check that This is the image representing the curves involved after applying π 1,1 : Proof.The idea of this proof is to take the equivalence π * 1,1 (D) ∼ • D+2 • F (1) in the weak degree 1 del Pezzo surface X, and follow the inverse process of blow ups and contractions to get an equivalence for π * a,b (D) in X a,b in terms of D and E. Let e k and f k be the coefficient of E (k) and F (k) , respectively, in the equivalence class of π * a,b (D).Define e to be the final coefficient of E. Notice that if k = 2n 0 , e = e i+son 0 + f j+sen 0 , otherwise e = e i+so n 0 +1 −1 + f j+sen 0 +1 .In addition, we know for the sequence of blowups that for n ≥ 0 Now let us rewrite b in terms of i, j, and j k using notation as in §4.1.
Here we have two different cases.
In both cases, as desired, we obtain e = 2b.Therefore, after all the contractions, we get π * a,b (D) ∼ D + 2b • E. To compute the self-intersection of D: Remark 4.9.Following a similar proof and taking into account that a = b + δ, it is easy to check that π * a,b (L) = L + aE, where L is the strict transform of L after the weighted blowup.

Proof of the Main Theorem
In the previous section, we prepared all the ingredients to prove the Main Theorem.Now, we divide this proof into two results: Theorem 5.1.Let X ⊂ P(1, 1, 1, 2) be a smooth del Pezzo surface of degree 2 and let p 0 = (x 0 , y 0 , z 0 , w 0 ) ∈ X be a closed point with w 0 ̸ = 0. Assume that there is a unique (-1)-curve L passing through the point p 0 .Then δ p 0 (X) ≤ 6 71 (11 + 8 √ 3).

LF
described as above.Then they do not intersect at the same point.Proof.In this proof, we have to take into account several things.Notice that in X, υ( • F (1) ) • υ( • D) = 3 and both of them contain the singular point coming from the contraction of • L.Then, after the blowup of the singular point at X, we recover • L again, and since • 2K X the intersections change as follows:

Corollary 4. 10 .
Let π a,b : X a,b → X be a (a, b)-weighted blowup of the point p 0 ∈ X, as described in Theorem 1.1.Let L be the unique (-1)-curve passing through the point p 0 and D the curve described above.Then,π * a,b (−K X ) ∼ 1 2 L + D + (a + 2b)E .