There are no Miyaoka–Kobayashi octics

Introduction

Let $C \subset\mathbb{P}^{2}$ be a reduced complex plane curve of degree $n$ having only ADE singularities. For a singular point $p$ of $C$ we denote by $\mu_p = \tau_p$ its Milnor (equal to its Tjurina) number, and by $G_p \subset SU(2)$ the binary polyhedral group of the ADE surface singularity $z^{2} = f_p(x,y)$ with $f_p = 0$ being a local equation of $C$ at $p$. Using Kähler–Einstein metrics on log canonical surfaces with branch loci, Kobayashi [13 Corollary 2] proved that for $n \ge7$ the following inequality holds:

Note that $\lvert\Gamma(p)\rvert= 2\lvert G_p\rvert$ in the notation of [13]. For $n$ even, (1) was obtained before by Hirzebruch and Ivinskis [11, 12] using the double cover of $\mathbb{P}^{2}$ branched along $C$. With the Miyaoka–Kobayashi numbers defined as $m(p)=3(\mu_p+1-1/\lvert G_p\rvert)$, see [7 Lemma 3.1], the inequality (1) reads as

and a curve $C$ of even degree $n\geq6$ is called a Miyaoka–Kobayashi curve, or an MK-curve, when equality holds, see [11, 12] and [7 Definition 2.4]. For $n\geq8$, the double cover of $\mathbb{P}^{2}$ branched along an MK-curve is a ball quotient surface, see [7 Section 2] for necessary details. Ivinskis proved in his master’s thesis [12] that there are exactly three types of MK-curves of degree 6 (see also [7 Theorem 3.9]). By contrast, no MK-curve of degree $n\geq8$ is known. In particular, the existence of MK-octics has remained open since Ivinskis raised the problem in his Bonn Diplomarbeit [12], supervised by Hirzebruch.

The inequality (1) is the case $D=\frac{1}{2}C$ of the orbifold Miyaoka–Yau inequality for the pair $(\mathbb{P}^{2},D)$, see [13 Theorem]. Langer [14] proved such an inequality for the pairs $(\mathbb{P}^{2},\alpha C)$ with any real $\alpha$ for which the pair is log canonical and $K_{\mathbb{P}^{2}}+\alpha C$ is pseudoeffective, see Theorem 2.1 below. This inequality was the main tool used in [7 Section 5]. For $\alpha=\frac{1}{2}$ it gives back (1), so an MK-curve gives an equality in Langer’s inequality at $\alpha=\frac{1}{2}$. When $n\geq8$, the value $\frac{1}{2}$ is an interior point of the interval of admissible values of $\alpha$, and both sides of Langer’s inequality depend smoothly on $\alpha$ near $\frac{1}{2}$. Hence the derivatives of both sides at $\alpha=\frac{1}{2}$ must be equal, and this gives a linear constraint on the singularities of an MK-curve. Recall that $\tau(C)=\sum_p\tau_p$ is the total Tjurina number of $C$.

Let us recall that the freeness defect $\nu(C)$ of $C$, which was introduced by Dimca in [5], is $0$ (resp. $1$) exactly when $C$ is free (resp. nearly free), and that $\nu(C) = 3m^2 - 3m + 1 - \tau(C)$ for any curve $C$ of degree $n = 2m$ with only ADE singularities, see [7 Section 6]. The following consequences are immediate.

Theorem 1.1, the MK condition and the upper bound $\tau(C) \le3m(m - 1) + 1$ reduce the possible singularities of an MK-curve of degree $8$ to 18 weak combinatorial types, see Proposition 4.1. A case by case analysis, based only on many beautiful classical facts on plane curves (the genus formula, Bézout’s theorem, the Plücker formulas and projective duality), shows that none of them can be realized. Hence we get our main somehow surprising result.

In degree 10 the same approach leads us to 284 numerical types, see Proposition 5.1. This is way too much for a case by case analysis and we do not know how to approach this problem effectively, but we can at least prove the following result.

Our proofs are based on small amount of symbolic computations, just at two places, namely for the enumeration of the numerical types in Propositions 4.1 and 5.1. In Appendix § 13 we give simple SINGULAR scripts performing these enumerations, together with a script recomputing the numerical data in Table 3. All the other arguments are done by hand.

	singularities	$\tau$	$\sum\delta_p$	$\sum I_p$	$\kappa$	decompositions	argument
O1	$4A_1+4D_4+D_5+2E_6$	37	25	18	5	none	Lemma 3.2
N1	$4A_1+A_2+3A_3+4D_4+D_5$	36	26	24	2	none	Lemma 3.2
N2	$10A_1+A_2+2D_4+2D_8$	36	27	26	1	none	Lemma 3.2
N3	$6A_1+4A_2+D_4+2D_5+D_8$	36	24	18	6	(5, 1, 1, 1)	Lemma 3.3
N4	$7A_1+A_2+2A_3+D_4+D_5+2D_6$	36	26	24	2	none	Lemma 3.2
N5	$6A_1+3A_2+A_3+A_4+D_4+D_5+E_8$	36	23	13	7	(6, 1, 1)	Lemma 3.3
N6	$7A_1+2A_2+A_5+4D_5$	36	24	18	6	(5, 1, 1, 1)	Lemma 3.3
N7	$2A_1+7A_2+4D_5$	36	21	10	11	(8), (7, 1)	Lemma 3.3
N8	$6A_1+3A_2+2A_3+D_5+D_6+E_7$	36	24	19	5	(5, 1, 1, 1)	Lemma 3.3
N9	$6A_1+4A_2+A_5+D_5+2E_6$	36	22	11	9	(7, 1)	Lemma 3.3
N10	$7A_1+4A_3+D_5+2E_6$	36	24	17	5	none	Lemma 3.2
N11	$A_1+9A_2+D_5+2E_6$	36	19	3	14	(8)	Lemma 3.3
T1	$13A_1+3A_2+A_{11}+D_5$	35	25	21	4	none	Lemma 3.2
T2	$14A_1+A_2+2A_7+D_5$	35	26	24	2	none	Lemma 3.2
T3	$11A_1+3A_3+2A_5+D_5$	35	26	25	1	(3, 1, 1, 1, 1, 1)	Proposition 4.2
T4	$6A_1+5A_2+3A_3+A_5+D_5$	35	23	17	6	(6, 1, 1), (5, 2, 1)	Proposition 4.3
T5	$7A_1+A_2+7A_3+D_5$	35	25	23	2	(4, 1, 1, 1, 1)	Proposition 4.4
T6	$A_1+10A_2+3A_3+D_5$	35	20	9	11	(8), (7, 1)	Proposition 4.5

Let us present the plan for our paper. In § 2 we explain how (1) fits into Langer’s inequality and prove Theorem 1.1 and Corollary 1.2. § 3 collects the classical facts on plane curves that we need, and § 4 and § 5 contain the proofs of Theorems 1.3 and 1.4. Appendix § 12 collects the local orbifold Euler numbers of the ADE singularities, and Appendix § 13 contains the computer scripts mentioned above. We work only in the projective setting over the complex numbers.

Kobayashi's inequality and Langer's inequality

Any ADE curve singularity $p$ has a weighted homogeneous normal form $f$, of weighted degree $1$ with respect to weights $w_1 \ge w_2$, see Table 1. Its log canonical threshold is

type	normal form	$\mu_p$	$|G_p|$	$(w_1,w_2)$	$\alpha_p$	$c_p$
$A_k$, $k \ge1$	$x^2+y^{k+1}$	$k$	$k+1$	$\left(\frac{1}{2},\frac{k+1}{k+1}\right)$	$\frac{1}{2}-\frac{1}{k+1}$	$3k$
$D_k$, $k \ge4$	$x^2y+y^{k-1}$	$k$	$4(k-2)$	$\left(\frac{k-2}{2(k-1)},\frac{1}{k-1}\right)$	$\frac{k-4}{2(k-3)}$	$\frac{3(k-1)(2k-3)}{2(k-2)}$
$E_6$	$x^3+y^4$	$6$	$24$	$\left(\frac{1}{3},\frac{1}{4}\right)$	$\frac{1}{12}$	$\frac{33}{2}$
$E_7$	$x^3+xy^3$	$7$	$48$	$\left(\frac{1}{3},\frac{2}{9}\right)$	$\frac{1}{9}$	$\frac{153}{8}$
$E_8$	$x^3+y^5$	$8$	$120$	$\left(\frac{1}{3},\frac{1}{5}\right)$	$\frac{2}{15}$	$\frac{87}{4}$

see [DS, Proposition 2.3].

We use Langer’s inequality [14 Corollary 5.2] for $X=\mathbb{P}^2$ and the $\mathbb{Q}$-divisor $\alpha C$, in the form [14 (11.1.1)], see also [7 Theorem 5.3]. Note that $K_{\mathbb{P}^2}+\alpha C$ is pseudoeffective exactly when $\alpha\ge\frac{3}{n}$.

For the ADE singularities we only need values of $e_{\mathrm{orb}}(p,\alpha)$ for $\alpha$ being close to $\frac{1}{2}$. The values for all $0\le\alpha\le\operatorname{lct}_p$ are collected in Table 4 in Appendix § 12.

Type	$e_{\mathrm{orb}}(0;\mathbb{C}^2,\alpha C)$
$A_1$	$(1-\alpha)^2,\quad0 \leq\alpha\leq1$
$A_{2k},\ k \geq1$	$\begin{cases} 1-2\alpha, & 0 \leq\alpha\leq\frac{2k-1}{4k+2}, \\ \frac{(2k+3-(4k+2)\alpha)^2}{8(2k+1)}, & \frac{2k-1}{4k+2}<\alpha\leq\frac{2k+3}{4k+2} \end{cases}$
$A_{2k+1},\ k \geq1$	$\begin{cases} 1-2\alpha, & 0 \leq\alpha\leq\frac{k}{2k+2}, \\ \frac{(k+2-(2k+2)\alpha)^2}{4(k+1)}, & \frac{k}{2k+2}<\alpha\leq\frac{k+2}{2k+2} \end{cases}$
$D_{2k},\ k \geq2$	$\begin{cases} (1-2\alpha)(1-\alpha), & 0 \leq\alpha\leq\frac{k-2}{2k-3}, \\ \frac{(k-(2k-1)\alpha)^2}{4(k-1)}, & \frac{k-2}{2k-3}<\alpha\leq\frac{k}{2k-1} \end{cases}$
$D_{2k+1},\ k \geq2$	$\begin{cases} (1-2\alpha)(1-\alpha), & 0 \leq\alpha\leq\frac{2k-3}{4k-4}, \\ \frac{(2k+1-4k\alpha)^2}{8(2k-1)}, & \frac{2k-3}{4k-4}<\alpha\leq\frac{2k+1}{4k} \end{cases}$
$E_6$	$\begin{cases} 1-3\alpha, & 0 \leq\alpha\leq\frac{1}{12}, \\ \frac{(7-12\alpha)^2}{48}, & \frac{1}{12}<\alpha\leq\frac{7}{12} \end{cases}$
$E_7$	$\begin{cases} 1-3\alpha, & 0 \leq\alpha\leq\frac{1}{9}, \\ \frac{(5-9\alpha)^2}{24}, & \frac{1}{9}<\alpha\leq\frac{5}{9} \end{cases}$
$E_8$	$\begin{cases} 1-3\alpha, & 0 \leq\alpha\leq\frac{2}{15}, \\ \frac{(8-15\alpha)^2}{60}, & \frac{2}{15}<\alpha\leq\frac{8}{15} \end{cases}$

Proof. This is [14 Theorem 9.4.2] applied to the minimal log resolution of $(\mathbb{C}^2,C)$, which is star-shaped for all ADE singularities, see also [1]. For $A_1$ and $D_4$ it also follows from [14 Theorem 8.3], and the cusp $A_2$ is treated in [14 11.2]. The value at $\alpha=\frac{1}{2}$ is obtained by substituting the weights from Table 1. It can also be seen directly: the double cover $Y$ of $\mathbb{C}^2$ branched along $C$ satisfies $K_Y=\pi^*(K_{\mathbb{C}^2}+\frac{1}{2}C)$ and has the quotient singularity $\mathbb{C}^2/G_p$ over $p$, so [14 (0.2.2) and Lemma 7.1] give $e_{\mathrm{orb}}\left(p,\frac{1}{2}\right)=\frac{1}{2}\cdot\frac{1}{|G_p|}$.

Let $\Phi(\alpha)$ be the right-hand side minus the left-hand side of the inequality in Theorem 2.1, so that $\Phi(\alpha)\ge0$ for $\frac{3}{n}\le\alpha\le\min_p\operatorname{lct}_p$. For $\alpha=\frac{1}{2}$, Lemma 2.2 gives

Hence, for $n\ge6$, Langer’s inequality for $\alpha=\frac{1}{2}$ is exactly Kobayashi’s inequality (1), and a curve $C$ of even degree is an MK-curve if and only if $\Phi\left(\frac{1}{2}\right)=0$.

Proof of Theorem 1.1. Since $n\ge8$, we have $\frac{3}{n}<\frac{1}{2}<\min_p\operatorname{lct}_p$, and by Lemma 2.2 the function $\Phi$ is a polynomial in $\alpha$ on a neighbourhood of $\frac{1}{2}$. As $\Phi\ge0$ and $\Phi\left(\frac{1}{2}\right)=0$, we get $\Phi'\left(\frac{1}{2}\right)=0$. The derivative of the right-hand side of Langer’s inequality at $\frac{1}{2}$ is $2n^2-3n$, and by Lemma 2.2 the derivative of the term corresponding to $p$ on the left-hand side is

with the values given in Table 1. □

Proof of Corollary 1.2. Since $d_p \ge0$, Theorem 1.1 gives $3\tau(C) \ge2n^2-3n=8m^2-6m$, which is (i). If all the singularities of $C$ are of type $A$, then $d_p=0$ for all $p$, and $3\tau(C)=8m^2-6m$ implies that $3$ divides $m$, hence also $n$. This proves (ii). For (iii), recall that $\tau(C) \le3m(m-1)+1$, see the proof of [7 Theorem 3.4], which relies on [10] and [8 Theorem 2.9]. Hence Theorem 1.1 gives $\sum_p d_p \le m^2-3m+3$, and the claim follows from $d_p \ge\frac{3}{4}$ for the points of type $D$ and $E$.

Remark 2.3. (1) For $n=6$ one has $\frac{3}{n}=\frac{1}{2}$, so the argument above only gives $\Phi'(\frac{1}{2})\ge0$, i.e. $\sum_p c_p\le54$. Nevertheless, the three types of MK-sextics described in [7 Definition 3.8] all satisfy the condition $\sum_p c_p=54=2n^2-3n$: $6c(A_1)+4c(A_3)=18+36,\qquad9c(A_2)=54,\qquad3c(A_1)+4c(D_4)=9+45$.

(2) In [7 Section 5], the cases (b) and (c) for octics with singularities of types $A_1,A_2,A_3,D_4$ were excluded using $\alpha=0.55$ and $\alpha=0.48$, respectively. This choice is explained by Theorem 1.1: one has $\Phi'(\frac{1}{2})=104-\sum_p c_p=-1$ in case (b) and $\Phi'(\frac{1}{2})=\frac{1}{2}$ in case (c).

(3) Corollary 1.2(ii) recovers [7 Corollaries 5.10 and 5.11] when $3$ does not divide $n$.

(4) Kobayashi’s inequality (1) holds also for odd $n\ge7$, and the proof of Theorem 1.1 shows that equality in (1) implies $\sum_p c_p=2n^2-3n$ for any $n\ge7$.

Some classical facts on plane curves

Let $C=C_1\cup\ldots\cup C_r$ be the decomposition of a reduced plane curve $C$ of degree $n$ into $r$ irreducible components. We set $d_i=\deg C_i$, we let $g_i$ be the geometric genus of $C_i$, and we put $p_a(e)=\frac{(e-1)(e-2)}{2}$. For a singular point $p$ of $C$ we denote by $\delta_p$ its $\delta$-invariant. We let $I_p$ be the sum of the intersection multiplicities of all pairs of distinct local branches of $C$ at $p$, and $\kappa_p$ the sum of $\operatorname{mult}(\beta)-1$ over all local branches $\beta$ of $C$ at $p$. Table 2 describes the local branches of the ADE singularities.

type	local branches	intersection numbers	$\delta_p$	$I_p$	$\kappa_p$
$A_{2k-1}$	two smooth branches	$k$	$k$	$k$	$0$
$A_{2k}$	one branch of multiplicity $2$	—	$k$	$0$	$1$
$D_{2k}$, $k\ge2$	smooth $L,B_1,B_2$	$L\cdot B_i=1,\ B_1\cdot B_2=k-1$	$k+1$	$k+1$	$0$
$D_{2k+1}$, $k\ge2$	smooth $L$, $S$ of type $A_{2k-2}$	$L\cdot S=2$	$k+1$	$2$	$1$
$E_6$	one branch of multiplicity $3$	—	$3$	$0$	$2$
$E_7$	smooth $L$, a cusp $S$ tangent to $L$	$L\cdot S=3$	$4$	$3$	$1$
$E_8$	one branch of multiplicity $3$	—	$4$	$0$	$2$

The following two lemmas are well-known.

Proof. For a point $p\in\mathbb{P}^2$, let $\delta(C_i,p)$ be the $\delta$-invariant of the germ $(C_i,p)$, so that $\delta(C_i,p)=0$ when $p\notin C_i$ or $C_i$ is smooth at $p$, and let $(C_i\cdot C_j)_p$ be the intersection multiplicity of $C_i$ and $C_j$ at $p$. We use the following three facts.

(a) The genus formula for the irreducible plane curve $C_i$ of degree $d_i$:

(b) Bézout’s theorem: for $1 \le i < j \le r$ one has

(c) For any point $p\in\mathbb{P}^2$ one has

To prove (c), consider first two plane curve germs $(X,p)$ and $(Y,p)$ without common irreducible components, and let $b(X,p)$ denote the number of local branches of $X$ at $p$. Milnor’s formula $2\delta(X,p)=\mu(X,p)+b(X,p)-1$, see [15 Theorem 10.5], together with the formula

see [Wa, Theorem 6.5.1], and the obvious equality $b(X\cup Y,p)=b(X,p)+b(Y,p)$, give

that is

Now let $p\in\mathbb{P}^2$, and assume that $C_1,\ldots,C_k$ are exactly the components of $C$ passing through $p$. All the terms in (2) involving some $C_i$ with $i>k$ vanish, so (2) is clear for $k\le1$. For $k\ge2$, it follows by induction on $k$, applying (3) to $X=C_1\cup\cdots\cup C_{k-1}$ and $Y=C_k$, and using the additivity $(X\cdot C_k)_p=\sum_{i<k}(C_i\cdot C_k)_p$ of intersection multiplicities.

Every singular point of a component $C_i$ and every point of $C_i\cap C_j$ with $i\ne j$ is a singular point of $C$, and both sides of (2) vanish when $p$ is a smooth point of $C$. Hence summing (2) over all $p\in\operatorname{Sing}(C)$ and using (a) and (b), we get

Finally, using $2p_a(e)=e^2-3e+2$ and $n=\sum_i d_i$, a direct computation gives

which proves the claimed formula. The last assertion follows from $g_i\ge0$. □

Proof. The intersection number $C_i\cdot C_j$ is the sum of the intersection multiplicities of the local branches of $C_i$ with the local branches of $C_j$, all located at singular points of $C$. The lower bound corresponds to the partition $n=(n-r+1)+\underbrace{1+\cdots+1}_{r-1}$.

For $n=8$ and $r=2,\ldots,7$ the lower bound in Lemma 3.2 is $7, 13, 18, 22, 25, 27$, and for $n=10$ and $r=2,\ldots,6$ it is $9, 17, 24, 30, 35$. Note also that lines and conics are smooth curves, hence they carry no singular local branch, and they carry at most one local branch at any given point.

For an irreducible curve $B\subset\mathbb{P}^2$ of degree $e\ge2$ and geometric genus $g$, let $\kappa(B)$ be the sum of $\operatorname{mult}(\beta)-1$ over all local branches $\beta$ of $B$. Then the class $e^\vee$ of $B$, i.e. the degree of the dual curve $B^\vee$, is given by

see [2 Section 9.1, formula (3) on p. 597]. This follows from the Riemann–Hurwitz formula for the projection of the normalization of $B$ from a general point, as in [2 Section 9.2, proof of Lemma 4], and it is equivalent to the formula $e^{\vee}=e(e-1)-\sum_p(\mu_p+\operatorname{mult}_p-1)$ used in [7 Section 4], see [4 (1.2.18)] and [2 Section 9.1, Theorem 2(i)]. Recall that $B^{\vee\vee}=B$, see [2 Section 6.2, p. 255], and that associating to a point of $B$ its tangent line gives a birational map $B\to B^\vee$, see [2 Section 9.1, p. 597]. Hence $B^\vee$ has degree $e^\vee$, geometric genus $g$ and class $e$. Applying (4) to $B^\vee$, as in [2 Section 9.1, formulae (4) and (5) on p. 597], we get the following.

Under the birational map $B\to B^\vee$, each local branch $\beta$ of $B$, with tangent line $T$, corresponds to a local branch $\beta^\vee$ of $B^\vee$ centered at the point $T^\vee$, and the multiplicity of $\beta^\vee$ is the class of $\beta$, see [2 Section 9.1, pp. 576–577]. We need the following facts about these dual branches.

Proof. For (a), choose affine coordinates with $p=(0,0)$ and $T=\{y=0\}$, and write $\beta$ as the graph $y=f(x)$, where $f$ has order $c$. The tangent line at the point $(x,f(x))$ is $y=ux-v$, with $u=f'(x)$ and $v=xf'(x)-f(x)$. Since $B\to B^\vee$ is birational, $x\mapsto(u(x),v(x))$ is a parametrization of $\beta^\vee$, and $u$, $v$ have orders $c-1$ and $c$ in $x$. Hence $\beta^\vee$ has multiplicity $c-1$. In the language of [2 Section 7.3, p. 288], an $r$-fold inflection point of $B$ gives a branch of $B^\vee$ of multiplicity $r+1$—this is the correspondence used in the proof of [2 Section 9.1, Theorem 2(ii)] on p. 597, and for $r=1$ it is the correspondence between simple inflection points and simple cusps [2 Section 9.1, p. 583].

For (b), note that the branch $\beta^\vee$ passes through $T^\vee$, and that its tangent line there is the line $p^\vee$ dual to the center $p$ of $\beta$. For a smooth branch $\beta$ this is the biduality statement in [2 Section 6.2, p. 255], and in general it follows by continuity, the tangent line of $\beta^\vee$ at $T^\vee$ being the limit of the lines dual to points of $\beta$ tending to $p$. Since $B\to B^\vee$ is birational, distinct branches of $B$ give distinct branches of $B^\vee$, which proves (b).

For (c), choose affine coordinates with $p=(0,0)$ and $T=\{y=0\}$, and write the two branches as graphs $y=f_j(x)=\sum_{i\ge2}a_{j,i}x^i$ with $a_{j,2}\ne0$, $j=1,2$. Since $k\ge3$, the coefficients $a_{1,i}$ and $a_{2,i}$ coincide for $i<k$, and $a_{1,k}\ne a_{2,k}$. The tangent line at the point $(x,f(x))$ is $y=ux-v$, with $u=f'(x)$ and $v=xf'(x)-f(x)$. Since $f''(0)\ne0$, the dual branch is the graph of the Legendre transform $v=F(u)$, which satisfies $F'(u)=x(u)$. Inverting $u=f'(x)$, one sees that, for $i\ge3$, the coefficient of $u^i$ in $F$ equals $-a_i/(2a_2)^i$ plus a polynomial in $a_2^{-1},a_3,\ldots,a_{i-1}$. Hence $F_1-F_2$ has order exactly $k$ in $u$, which is our claim.

Finally, if $\beta_1,\ldots,\beta_s$ are the local branches of a curve at a point $q$, then $\delta_q=\sum_j\delta(\beta_j)+\sum_{i<j}(\beta_i\cdot\beta_j)_q$ and $\delta(\beta_j)\ge\operatorname{mult}(\beta_j)-1$. Summing over all the points of $B^\vee$ we get

Proof of Theorem 1.3

For $n=8$, the MK condition reads $\sum_p\left(\mu_p+1-\frac{1}{\lvert G_p\rvert}\right)=\frac{136}{3}$, Theorem 1.1 reads $3\tau(C)-\sum_p d_p=104$, and we have $\tau(C)\le37$.

Proof. By Corollary 1.2, the singularities of type $D$ and $E$ satisfy $\sum_p d_p\le7$. For each such choice, $\tau(C)$ is determined by Theorem 1.1, and the singularities of type $A$ form a partition of the remaining Tjurina number satisfying one linear equation. The resulting finite list is obtained by the script in Appendix § 13.§ 1. □

The numbers $\sum_p\delta_p$, $\sum_p I_p$ and $\kappa=\sum_p\kappa_p$ in Table 3 are obtained by adding the entries of Table 2. By Lemma 3.1,

and the decompositions $8=d_1+\cdots+d_r$ compatible with Lemma 3.2 are easily listed by hand. For the sanity check, the script in Appendix § 13.§ 2 recomputes these data. Our aim not is to exclude the 18 types and we will do it in three steps.

Types excluded by numerics

For the types O1, N1, N2, N4, N10, T1 and T2, the number of irreducible components forced by Lemma 3.1 is incompatible with Lemma 3.2. For instance, for O1 we have $\sum_p\delta_p=25$, hence $r\ge5$ and $\sum_{i<j}d_i d_j\ge22$, while $\sum_p I_p=18$, a contradiction. The other cases can be treated in a similar way, namely

• N1, N4: $r\ge6$, so $\sum_{i<j}d_i d_j\ge25>24$;

• N2: $r \ge7$, so $\sum_{i<j} d_i d_j \ge27 > 26$;

• N10: $r \ge4$, so $\sum_{i<j} d_i d_j \ge18 > 17$;

• T1: $r \ge5$, so $\sum_{i<j} d_i d_j \ge22 > 21$;

• T2: $r \ge6$, so $\sum_{i<j} d_i d_j \ge25 > 24$;

Types excluded by the Plücker lemma

For the types N3, N5–N9 and N11, Lemmas 3.1 and 3.2 leave only the decompositions listed in Table 3, together with the value of $\sum_i g_i$. In each of them there is exactly one component $B$ of degree $e \ge3$, the other components are lines, so $g(B) = \sum_i g_i$, and $B$ contains all the singular local branches of $C$, so $\kappa(B) = \kappa$. The class $e^\vee$ given by (4) then violates Lemma 3.3:

• N3 and N6: $B$ is a rational quintic with $\kappa(B) = 6$, so $e^\vee= 2$ and $\kappa(B^\vee) = -3$;

• N5: $B$ is a rational sextic with $\kappa(B) = 7$, so $e^\vee= 3$ and $\kappa(B^\vee) = -2$;

• N7: $B$ is either a rational octic or a septic of genus 1, with $\kappa(B) = 11$, so $e^\vee= 3$ and $\kappa(B^\vee) < 0$;

• N8: $B$ is a rational quintic with $\kappa(B) = 5$, so $e^\vee= 3$ and $\kappa(B^\vee) = -1$;

• N9: $B$ is a rational septic with $\kappa(B) = 9$, so $e^\vee= 3$ and $\kappa(B^\vee) = -3$;

• N11: here $C$ is irreducible of genus 2 and $\kappa(C) = 14$, so $e^\vee= 4$ and $C^\vee$ is a quartic of genus 2, so that $\delta(C^\vee) = 1 < 2 = \kappa(C^\vee)$.

The four remaining types

Proof. Here $r-\sum g_i=5$ and $\sum_p I_p=23$, so Lemma 3.2 gives the decomposition $(4,1,1,1,1)$, all components rational, and $\sum_{i<j}d_i d_j=22$. Hence the intersection multiplicities between branches lying on the same component add up to 1. The quartic $Q$ contains the $A_2$ point and the singular branch of $D_5$, and the remaining unit is a node of $Q$. Thus $Q$ is a rational quartic with two cusps and one node, and $Q$ is smooth at all the other points of $C$. Then $Q$ has class 4, and $Q^\vee$ is a rational quartic with $\kappa(Q^\vee)=2$ and $\delta(Q^\vee)=3$.

No singularity in T5 has three local branches. Hence the six intersection points of the four lines are $A_1$ points not on $Q$, and together with the node of $Q$ they account for all seven $A_1$ points. It follows that

comes from the seven $A_3$ points, each being a tangency of a line with $Q$, and from the $D_5$ point, where a line passes through a cusp of $Q$. Since each line meets $Q$ with multiplicity 4, three of the four lines are bitangent to $Q$. By Lemma 3.4(b) this gives three distinct points of $Q^\vee$ with at least two branches, and (5) yields $\delta(Q^\vee)\ge2+3=5>3$, a contradiction. □

Proposition 4.5. The type T6 = $A_1+10A_2+3A_3+D_5$ is not realized by a plane octic.

Proof. Here $r=\sum g_i$ and $\sum_p I_p=9$, so $r\le2$. Hence either $C$ is irreducible of genus 1, or $C=S\cup L$ with $S$ a septic of genus 2 and $L$ a line. In both cases the component of degree $\ge7$ has $\kappa=11$.

In the second case, $S$ has class $14+4-2-11=5$, and $S^\vee$ is a quintic of genus 2 with $\kappa(S^\vee)=10+4-2-7=5>4=\delta(S^\vee)$, contradicting Lemma 3.3.

In the first case, $C$ has class $16+2-2-11=5$, and $C^\vee$ is a quintic of genus 1, so $\delta(C^\vee)=5$. Let $T_1,T_2,T_3$ be the tangent lines at the three tacnodes of $C$. By Lemma 3.4(b), each $T_j$ is a point of $C^\vee$ carrying two branches with intersection multiplicity at least 2. If the $T_j$ are distinct, (5) gives $\delta(C^\vee)\ge6$. If two of them coincide, the corresponding point of $C^\vee$ has at least four branches, and (5) gives $\delta(C^\vee)\ge2+2+4=8$. In both cases we get a contradiction. □

This completes the proof of Theorem 1.3.

Remark 4.6. Proposition 4.1 shows that an MK-octic would have a singularity of type $D_5$ or $D_8$. This already recovers, in degree 8, the results [7 Theorem 5.7, Corollaries 5.10 and 5.11, Theorems 5.12 and 5.16]. Note also that the numerical types of free and nearly free MK-octics are excluded by the simplest arguments above, while in degree 6 one MK-sextic is free and the other two are nearly free, see [7 Example 6.4].

Proof of Theorem 1.4

For $n=10$, the MK condition reads $\sum_p\left(\mu_p+1-\frac{1}{|G_p|}\right)=\frac{220}{3}$, Theorem 1.1 reads $3\tau(C)-\sum_p d_p=170$, and we have $\tau(C)\le61$.

Proof. The proof is the same as for Proposition 4.1, see the script in Appendix § 13.§ 1. □

Proof of Theorem 1.4. By Corollary 1.2 and Proposition 5.1 it is enough to exclude the types F1 and F2. We use the notation of § 3, and the data can be checked with the script in Appendix § 13.§ 2.

For F1 one has $\sum_p\delta_p=38$ and $\sum_p I_p=23$, hence $r-\sum g_i=3$. Lemma 3.2 leaves only two decompositions $(8,1,1)$ and $(7,2,1)$, with all components rational. The component $B$ of degree $e\ge7$ contains all singular branches, so $\kappa(B)=4\cdot2+2+1=11$. If $e=8$, then $e^\vee=3$ and $\kappa(B^\vee)=-4$. If $e=7$, then $e^\vee=1$. Both contradict Lemma 3.3.

For F2 one has $\sum_p\delta_p=39$ and $\sum_p I_p=31$, hence $r-\sum g_i=4$, and $\kappa=8$. Lemma 3.2 leaves three decompositions:

• $(7,1,1,1)$ with all components rational: here $e^{\vee}=4$ and $\kappa(B^{\vee})=-1$, which contradicts Lemma 3.3;

• $(6,2,1,1)$ with all components rational: here $e^{\vee}=2$ and $\kappa(B^{\vee})=-4$, again a contradiction;

• $(6,1,1,1,1)$, where the sextic $B$ has genus 1.

In the last case $B$ has class 4, so $B^{\vee}$ is a quartic of genus 1 with $\delta(B^{\vee})=2$ and $\kappa(B^{\vee})=2$. By (5), no point of $B^{\vee}$ can have two or more local branches.

On the other hand, $\delta(B)=p_a(6)-1=9$, and the eight singular branches on $B$ (the $A_2$ point, the singular branch of $D_5$ and the cusps of the six $E_7$ points) contribute 8 to $\delta(B)$. If both branches of an $E_7$ point lay on $B$, they would contribute 3 more, which is impossible. Hence for each $E_7$ point, the smooth branch lies on a line, which is therefore the tangent line to $B$ at the corresponding cusp. Each such line meets $B$ with multiplicity 3 at this cusp, so a line can be the cuspidal tangent at no more than two of these cusps. Since there are six $E_7$ points and four lines, at least one line $L$ is tangent to $B$ at two distinct cusps. By Lemma 3.4(b), $L^{\vee}$ is a point of $B^{\vee}$ with at least two branches, a contradiction. □

Remark 5.2. A systematic, computer aided application of the arguments of § 3 and § 4 to the remaining 282 numerical types of degree 10 excludes a large part of them, but by no means all of them. In the remaining cases the curve typically has an irreducible component of degree at least 7, for which the counting arguments of § 4 give no contradiction. We do not pursue this here due to lack of efficient techniques.

We would like to conclude our paper by the following difficult problem.

Question 5.3. Is there a Miyaoka–Kobayashi curve of degree 10? Is there a nearly free one?

Acknowledgements

We would like to remember Wolfgang Ebeling, a friend of both authors and an exceptionally kind and warm-hearted person. It was Wolfgang who first introduced Piotr Pokora to MK-curves: after a discussion in Hannover, he handed him a copy of Ivinskis’ Diplomarbeit, a small gesture, entirely typical of his generosity, that some years later became the starting point of our interest in these curves. Wolfgang was a doctoral student of Egbert Brieskorn, and in that same spirit we have drawn many of the results used here from the wonderful book Plane Algebraic Curves by Brieskorn and Knörrer.

Funding

Piotr Pokora is supported by the National Science Centre (Poland) Sonata Bis Grant 2023/50/E/ST1/00025. For the purpose of Open Access, the author has applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission.

AI Declaration

All mathematical results in this paper were obtained by the authors. We used Claude (Anthropic) only to consolidate scattered symbolic computations in SINGULAR into two scripts that readers can run on their own computers.

Data Availability Statement

The scripts supporting the computations in this paper are available in Appendix § 13.

Conflict of Interest

The authors declare that they have no conflicts of interest related to this manuscript.

References

Alexandru Dimca: Université Côte d’Azur, CNRS, LJAD, France

Simion Stoilow Institute of Mathematics, Bucharest, Romania Email address: [email protected]

Piotr Pokora: Department of Mathematics, University of the National Education Commission Krakow, Podchorążych 2, PL-30-084 Kraków, Poland Email address: [email protected]

Local orbifold Euler numbers for ADE singularities

Table 4 lists the local orbifold Euler numbers of the ADE singularities for $0 \leq\alpha\leq\operatorname{lct}_p$. For $\alpha_p \leq\alpha\leq\operatorname{lct}_p$, with $\alpha_p$ as in Table 1, the entries agree with the formula in Lemma 2.2.

Singular scripts

The following scripts were run with SINGULAR 4.3.2 [3]. In both scripts a singularity type is coded by an integer: $A_k$ by $k$, $D_k$ by $100+k$ and $E_k$ by $200+k$.

The numerical types

This script proves Propositions 4.1 and 5.1. It lists all the numerical types for $n=8$, and for $n=10$ it prints their number for each value of $\tau$ together with the two types with $\tau=61$.

// mk_candidates.sing -- Propositions 4.1 and 5.1.
// Numerical types of MK-curves of degree n = 2m, i.e. multisets of ADE types with
// (MK) sum_p (mu_p + 1 - 1/|G_p|) = n(5n-6)/6, see (1),
// (DER) 3 tau - sum_p d_p = 2n^2 - 3n, see Theorem 1.1,
// (TAU) tau <= 3m(m-1) + 1.
// A type is coded by an integer: A_k = k, D_k = 100 + k, E_k = 200 + k.
ring R = 0, x, dp;                         // only used for rational numbers
proc name(int c)
{
    if (c < 100) { return("A" + string(c)); }
    if (c < 200) { return("D" + string(c - 100)); }
    return("E" + string(c - 200));
}
// prints a type, e.g. 4D4+D5+2E6+4A1, from a list of codes (equal codes adjacent)
proc show(list v)
{
    string s; int i, j;
    for (i = 1; i <= size(v); i = j)
    {
        for (j = i + 1; j <= size(v); j++) { if (v[j] != v[i]) { break; } }

if (s != "") { s = s + "+"; }
if (j > i + 1) { s = s + string(j - i); }
s = s + name(v[i]);
}
" " + s;
}
// global data: the D and E types with mu_p, mu_p + 1 - 1/|G_p| and d_p (Table 1, Theorem 1.1)
list DE, MU, MK, DP; int CNT; int SHOW;
// A part: partitions of t into parts k <= p with sum_k k/(k+1) = q.
// Each such part contributes between k/(p+1) and k/2 to q.
proc aRec(int t, int p, number q, list v)
{
    if (t == 0) { if (q == 0) { CNT++; if (SHOW) { show(v); } } return(); }
    if (q*(p + 1) < t || 2*q > t) { return(); }
    int k;
    for (k = p; k >= 1; k--) { if (k <= t) { aRec(t - k, k, q - number(k)/(k + 1), v + list(k)); } }
}
// D,E part: d = remaining sum of d_p, t = remaining tau, q = remaining left-hand side of (MK).
// The A part needs q - t <= t/2, and adding a D or E point only makes this worse.
proc deRec(int i, number d, int t, number q, list v)
{
    if (2*q > 3*t) { return(); }
    if (d == 0) { aRec(t, t, q - t, v); return(); }
    int j;
    for (j = i; j <= size(DE); j++)
    {
        if (DP[j] <= d && MU[j] <= t) { deRec(j, d - DP[j], t - MU[j], q - MK[j], v + list(DE[j])); }
    }
}
// prints the types with tau = 3m(m-1)+1, and all types if all = 1
proc candidates(int n, int all)
{
    int m = n div 2; int tmax = 3*m*(m - 1) + 1; int k, tau;
    DE = list(); MU = list(); MK = list(); DP = list();
    for (k = 4; k <= tmax; k++)
    {
        DE = DE + list(100 + k); MU = MU + list(k);
        MK = MK + list(k + 1 - 1/number(4*k - 8)); DP = DP + list(number(3*k - 9)/(2*k - 4));
    }
    DE = DE + list(206, 207, 208); MU = MU + list(6, 7, 8);
    MK = MK + list(7 - 1/24, 8 - 1/48, 9 - 1/120); DP = DP + list(3/2, 15/8, 9/4);
    for (tau = tmax; 3*tau >= 2*n^2 - 3*n; tau--)
    {
        CNT = 0; SHOW = all || (tau == tmax);
        deRec(1, 3*tau - 2*n^2 + 3*n, tau, number(n*(5*n - 6))/6, list());
        "n = " + string(n) + ", tau = " + string(tau) + ", nu = " + string(tmax - tau)
             + ": " + string(CNT) + " type(s)";
    }
}
candidates(8, 1);        // Proposition 4.1: 1, 11 and 6 types with tau = 37, 36, 35
candidates(10, 0);       // Proposition 5.1: 2, 48, 108, 96, 30 types with tau = 61, ..., 57
quit;

The output for $n = 8$ consists of the 18 types in Table 3, listed in the same order: there are 1, 11 and 6 types with $\tau= 37, 36, 35$, respectively. For $n = 10$ it gives 284 types: there are 2, 48, 108, 96 and 30 types with $\tau= 61, 60, 59, 58, 57$, respectively, and the two types with $\tau= 61$ are F1 and F2.

Numerical data for Sections 4 and 5

For each type in Table 3 and for the types F1, F2 of Proposition 5.1, this script computes $\sum_p \delta_p$, $\sum_p I_p$ and $\kappa$, lists the decompositions allowed by Lemmas 3.1 and 3.2, and applies the Plücker test of Lemma 3.3 when there is a single component of degree at least 3. The test fails in all cases except T3, T4 with $(6,1,1)$, T5, T6 with (8) and F2 with $(6,1,1,1)$. These are exactly the cases treated by hand in Propositions 4.2–4.5 and in the proof of Theorem 1.4.

// mk_geometry.sing -- the data in Table 3 and in the proof of Theorem 1.4.
// A type is coded by an integer: A_k = k, D_k = 100 + k, E_k = 200 + k, and a
// configuration by an intvec (code_1, mult_1, code_2, mult_2, ...).
// For a configuration on a curve of degree n we print sum delta_p, sum I_p and
// kappa, the partitions of n allowed by Lemmas 3.1 and 3.2, and exactly one
// part is at least 3, the Pluecker test of Lemma 3.3 for that component.
ring R = 0, x, dp;
proc pa(int e) { return(((e - 1)*(e - 2)) div 2); }
// (delta_p, I_p, kappa_p), see Table 2
proc data(int c)
{
    int k = c mod 100;
    if (c < 100)

{
    if (k mod 2 == 1) { return(intvec((k + 1) div 2, (k + 1) div 2, 0)); }
    return(intvec(k div 2, 0, 1));
}
if (c < 200)
{
    if (k mod 2 == 0) { return(intvec(k div 2 + 1, k div 2 + 1, 0)); }
    return(intvec(k div 2 + 1, 2, 1));
}
if (k == 6) { return(intvec(3, 0, 2)); }
if (k == 7) { return(intvec(4, 3, 1)); }
return(intvec(4, 0, 2));
// the partitions of n with parts <= mx
proc partitions(int n, int mx)
{
    list P, Q; int p, j;
    if (n == 0) { return(list(intvec(0))); }
    for (p = n; p >= 1; p--)
    {
        if (p <= mx)
        {
            Q = partitions(n - p, p);
            for (j = 1; j <= size(Q); j++) { P = insert(P, intvec(p, Q[j]), size(P)); }
        }
    }
    return(P);
}
proc analyse(string name, int n, intvec cfg)
{
    int D, I, K, i, j, r, G, s2, cross, nbig, e, cl, kd, dd, ok, found;
    intvec d, p; list P; string out;
    for (i = 1; i < size(cfg); i = i + 2)
    {
        d = data(cfg[i]);
        D = D + cfg[i+1]*d[1]; I = I + cfg[i+1]*d[2]; K = K + cfg[i+1]*d[3];
    }
    name + ": sum delta = " + string(D) + ", sum I = " + string(I) + ", kappa = "
    + string(K) + ", r - sum g = " + string(D - pa(n) + 1);
    P = partitions(n, n);
    for (i = 1; i <= size(P); i++)
    {
        // drop the trailing 0 of the partition
        p = P[i]; p = p[1..size(p) - 1];
        r = size(p); G = r - (D - pa(n) + 1);
        s2 = 0;
        for (j = 1; j <= r; j++) { s2 = s2 + p[j]^2; }
        cross = (n^2 - s2) div 2;
        if (G >= 0 && cross <= I)
        {
            found = 1;
            out = " (" + string(p) + "): sum g = " + string(G);
            out = out + ", sum d_i d_j = " + string(cross);
            nbig = 0;
            for (j = 1; j <= r; j++) { if (p[j] >= 3) { nbig++; e = p[j]; } }
            if (nbig == 1)
            {
                // class of the component of degree e >= 3 and kappa, delta of its dual
                cl = 2*e + 2*G - 2 - K;
                out = out + "; degree " + string(e) + ", genus " + string(G);
                out = out + ", class " + string(cl);
                ok = (cl >= 2);
                if (ok)
                {
                    kd = 2*cl + 2*G - 2 - e; dd = pa(cl) - G;
                    out = out + ", dual: delta = " + string(dd) + ", kappa = " + string(kd);
                    ok = (kd >= 0) && (kd <= dd);
                    if (ok) { out = out + " -> Pluecker test passed"; }
                    else { out = out + " -> Pluecker test FAILS"; }
                }
            }
            if (nbig > 1) { out = out + "; several components of degree >= 3"; }
            out;
        }
    }
    if (!found) { " no partition satisfies Lemma 3.2"; }
}
"=== degree 8, tau = 37 ===";
analyse("O1 4A1+4D4+D5+2E6", 8, intvec(1,4, 104,4, 105,1, 206,2));

"=== degree 8, tau = 36 ===";
analyse("N1 4A1+A2+3A3+4D4+D5", 8, intvec(1,4, 2,1, 3,3, 104,4, 105,1));
analyse("N2 10A1+A2+2D4+2D8", 8, intvec(1,10, 2,1, 104,2, 108,2));
analyse("N3 6A1+4A2+D4+2D5+D8", 8, intvec(1,6, 2,4, 104,1, 105,2, 108,1));
analyse("N4 7A1+A2+2A3+D4+D5+2D6", 8, intvec(1,7, 2,1, 3,2, 104,1, 105,1, 106,2));
analyse("N5 6A1+3A2+A3+A4+D4+D5+E8", 8, intvec(1,6, 2,3, 3,1, 4,1, 104,1, 105,1, 208,1));
analyse("N6 7A1+2A2+A5+4D5", 8, intvec(1,7, 2,2, 5,1, 105,4));
analyse("N7 2A1+7A2+4D5", 8, intvec(1,2, 2,7, 105,4));
analyse("N8 6A1+3A2+2A3+D5+D6+E7", 8, intvec(1,6, 2,3, 3,2, 105,1, 106,1, 207,1));
analyse("N9 6A1+4A2+A5+D5+2E6", 8, intvec(1,6, 2,4, 5,1, 105,1, 206,2));
analyse("N10 7A1+4A3+D5+2E6", 8, intvec(1,7, 3,4, 105,1, 206,2));
analyse("N11 A1+9A2+D5+2E6", 8, intvec(1,1, 2,9, 105,1, 206,2));
"=== degree 8, tau = 35 ===";
analyse("T1 13A1+3A2+A11+D5", 8, intvec(1,13, 2,3, 11,1, 105,1));
analyse("T2 14A1+A2+2A7+D5", 8, intvec(1,14, 2,1, 7,2, 105,1));
analyse("T3 11A1+3A3+2A5+D5", 8, intvec(1,11, 3,3, 5,2, 105,1));
analyse("T4 6A1+5A2+3A3+A5+D5", 8, intvec(1,6, 2,5, 3,3, 5,1, 105,1));
analyse("T5 7A1+A2+7A3+D5", 8, intvec(1,7, 2,1, 3,7, 105,1));
analyse("T6 A1+10A2+3A3+D5", 8, intvec(1,1, 2,10, 3,3, 105,1));
"=== degree 10, tau = 61 ===";
analyse("F1 6A1+3D4+D5+4E6+2E7", 10, intvec(1,6, 104,3, 105,1, 206,4, 207,2));
analyse("F2 8A1+A2+D4+D5+6E7", 10, intvec(1,8, 2,1, 104,1, 105,1, 207,6));
quit;