On rigid plane curves
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Abstract
We exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
Keywords
Plane curves Rigid curves Equisingular families of curvesMathematics Subject Classification
14H10 14H20 14H501 Introduction
1.1 Background and motivation
We work over the complex field, though most of results can be stated over any algebraically closed field of characteristic zero.
The space \({\mathscr {C}}_d\) of plane curves of degree d can be identified with Open image in new window . It has a natural equisingular stratification with the strata determined by the collection of degrees and multiplicities of irreducible components and by the collection of topological singularity types of the considered curves (see [10, 11]; below, the strata will be called the families of equisingular curves). Properties of this stratification have been studied by algebraic geometers since 19th century, attracting attention of leading experts like Zeuthen, Severi, Segre, Zariski and others (see, for example, [2] for a modern survey in this area).
In this paper we focus on the minimal equisingular families, that is, those which are formed by reduced curves and contain only finitely many orbits of the action of the group of projective transformations of the plane Open image in new window . The curves belonging to these families are called rigid curves; the corresponding families we also call rigid (see Definition 1.2 below). The study of rigid curves is motivated by their appearance in several important problems. First of all, finite coverings of the projective plane branched along rigid curves are used to obtain examples of rigid, so called, Miyaoka–Yau surfaces (see, for example, [3, 5]). Next (cf. [6]), celebrated Belyi’s Theorem [1] says that each projective curve defined over \(\overline{\mathbb {Q}}\) can be represented as a finite covering of \(\mathbb {P}^1\) branched at three points. Note that any three points in \(\mathbb {P}^1\) are rigid in the sense of definition given below. Therefore we can hope that for any field F of transcendence degree two over \(\overline{\mathbb {Q}}\) there are a projective model X defined over \(\overline{\mathbb {Q}}\) with the field of rational functions \(\overline{\mathbb {Q}}(X)\simeq F\) and a finite morphism Open image in new window branched along a rigid plane curve. There are interesting relations to the geometry of line arrangements (see [3]) and to rational cuspidal curves (see [9]).
The goal of our note is to exhibit examples of rigid curves of any degree and any genus, and rigid families covered by arbitrarily many orbits of the Open image in new window action.
1.2 Definitions and main results
Throughout the paper we consider isolated plane curve singular points up to topological equivalence, briefly calling any class of topologically equivalent singular points a singularity type. Given a singularity type S, the number of irreducible components of singular curve germs of type S is an invariant, which we denote \(m_S\). Cardinality of a finite set F will be denoted by F.
1.2.1 Irreducible rigid curves
Such a family is a locally closed union of quasiprojective subvarieties of \({\mathscr {C}}_d\) (cf. [2]). It is, of course, invariant with respect to the action of Open image in new window , and hence consists of entire orbits of the Open image in new window action.
Definition 1.1
We say that a nonempty equisingular family \(V(d;g;{\mathbf S})\) is k rigid if it is the union of k distinct orbits of the Open image in new window action in \({\mathscr {C}}_d\) for some \(k\in {\mathbb {N}}\). If \(k=1\) then we say that V is strictly rigid. The curves belonging to a krigid family of plane curves are called rigid.
1.2.2 Reducible rigid curves
Considering reduced, irreducible curves, we, first, introduce families of reducible curves with numbered components, then identify families obtained from each other by permutation of components.
Let \(\mathbf{d}=(d_1,\dots , d_N)\) and \(\mathbf{g}=(g_1,\dots , g_N)\) be two collections of integers, \(d_i\ge 1\) and \(g_i\ge 0\) for \(i=1,\dots , N\). To encode the distribution of singularity types among components and the distribution of local branches centered at singular points that are intersection points of components, we do the following. For a fixed singularity type S and fixed N denote by \(\mathbf{J}_S=\{ J_{S,k}\}\) the set of all nonempty subsets \(J_{S,k}\) of \(\{1,2,\dots ,N\}\), \(1\le k\le \sum _{j=1}^{m_S} \left( {\begin{array}{c}N\\ j\end{array}}\right) \), such that \(J_{S,k}\le m_{S}\). Let \(V\bigl (\mathbf{d};\mathbf{g}; \sum _{\{ S_j\}}\sum _{\{ J_{S_j,k}\}}n_{J_{S_j,k}}S_j\bigr )\) be the family of plane reduced curves \(\overline{C}=C_1\cup \dots \cup C_N\subset {\mathbb {P}}^2\) such that \(C_i\) are irreducible curves of degree \(\deg C_i=d_i\) and genus \(g_i\), and for each type \(S_j\) of plane singularities the intersection \(\bigcap _{i\in J_{S_j,k}}C_i\) contains exactly \(n_{J_{S_j,k}}\) singular points of \(\overline{C}\) of the type \(S_j\) which do not lie in \(C_l\) for \(l\not \in J_{S_j,k}\). This is a locally closed union of quasiprojective subvarieties of \({\mathscr {C}}_d\) (cf. [2]).
The sum \(\mathbf{S}=\sum _{\{ S_j\}}\sum _{\{ J_{S_j,k}\}}n_{J_{S_j,k}}S_j\) is called the singularity type of the curves \(\overline{C}\in V(\mathbf{d};\mathbf{g};\mathbf{S})\). We identify the families \(V(\mathbf{d};\mathbf{g};\mathbf{S})\) obtained by permutations of the curves \(C_1,\dots ,C_N\) and compatible permutations of \(\mathbf{S}\).
A singularity type \(\mathbf{S}\) splits into two parts, \(\mathbf{S}=\mathbf{S}^\mathrm{ess}+\mathbf{S}^\mathrm{non\text {}ess}\), as follows. For fixed \(\mathbf{d}\) and \(\mathbf{g}\), we say that \(\mathbf{S}^\mathrm{ess}\) is an essential part of the singularity type \(\mathbf{S}\) (and resp. \(\mathbf{S}^\mathrm{non\text {}ess}\) is a nonessential part of the singularity type \(\mathbf{S}\)) if the family \(V(\mathbf{d;\mathbf g; \mathbf S})\) is determined uniquely by \(\mathbf{d},\mathbf{g}\), and the property that the curves \(\overline{C}\) have the singularities \(\mathbf{S}^\mathrm{ess}\) among all singularities of \(\overline{C}\). If \(\mathbf{S}^\mathrm{ess}\) is an essential part of a singularity type \(\mathbf{S}\), then we will use notation \(V(\mathbf{d;\mathbf g;\mathbf S}^\mathrm{ess}+\cdots )\) to denote the family \(V(\mathbf{d;\mathbf g;\mathbf S})\).
Definition 1.2
We say that a family \(V=V(\mathbf{d;\mathbf g; \mathbf S})\) is k rigid if it is the union of k distinct orbits of the Open image in new window action in \({\mathscr {C}}_d\) for some \(k\in {\mathbb {N}}\). If \(k=1\) then we say that V is strictly rigid. The curves belonging to a krigid family of plane curves are called rigid.
1.2.3 Main results

in Theorem 2.1, the complete list of rigid curves of degree \(\le 4\) is given;

in Theorem 3.1, we give an infinite series of examples of strictly rigid families of irreducible rational curves \(V(d;0;\mathbf{S})\);

in Theorem 4.1, for each \(g\ge 1\), we prove the existence of strictly rigid irreducible plane curves of genus g;

examples of irreducible 2rigid families of irreducible curves are given in Theorems 5.1 and 5.2, and Theorem 5.3 provides examples of krigid families \(V(\mathbf{d};\mathbf{g};\mathbf{S})\) consisting of k irreducible components for each \(k\in {\mathbb {N}}\).
Question 1.3
Do there exist irreducible krigid families \(V(\mathbf{d};\mathbf{g};\mathbf{S})\) with \(k> 2\)?
Question 1.4
Do there exist irreducible 2rigid families \(V(d;g;\mathbf{S})\) with \(g\ge 1\)?

\(T_{m,n}\), \(2\le m\le n\), is the type of singularity given by the equation \(x^m+y^{n}=0\); but if \(m=2\) then a singularity of type \(T_{2,n}\), as usual, will be denoted by \(A_{n1}\) and the singularities of types \(T_{m,m}\) will be called simple.

\(T^m_{m,n}\), \(2\le m< n\), is the type of singularity given by the equation Open image in new window .

\(T^n_{m,n}\), \(2\le m< n\), is the type of singularity given by the equation Open image in new window .

\(T^{m,n}_{m,n}\), \(1\le m< n\), is the type of singularity given by the equation Open image in new window .
2 Rigid curves of small degree
In the following theorem, we provide the complete list of rigid reduced curves of degree \(\le 4\).
Theorem 2.1
 (I)
strongly rigid families:
 (I\(_1)\)

\(V(1;0;\varnothing )\);
 (I\(_2)\)

\(V\bigl ((1,1);(0,0);A_1\bigr )\);
 (I\(_3)\)

\(V(2;0;\varnothing )\);
 (I\(_4)\)

\(V\bigl ((1,1,1);(0,0,0);3A_1\bigr )\);
 (I\(_5)\)

\(V\bigl ((1,1,1);(0,0,0);T_{3,3}\bigr )\);
 (I\(_6)\)

\(V\bigl ((2,1);(0,0);2A_1\bigr )\);
 (I\(_7)\)

\(V\bigl ((2,1);(0,0);A_3\bigr )\);
 (I\(_8)\)

\(V(3;0;A_1)\);
 (I\(_9)\)

\(V(3;0;A_2)\);
 (I\(_{10})\)

\(V\bigl ((1,1,1,1);(0,0,0,0);6A_1\bigr )\);
 (I\(_{11})\)
 (I\(_{12})\)
 (I\(_{13})\)
 (I\(_{14})\)
 (I\(_{15})\)
 (I\(_{16})\)
 (I\(_{17})\)
 (I\(_{18})\)

\(V\bigl ((3,1);(0,0);T_{2,4}^2\bigr )\);
 (I\(_{19})\)

\(V\bigl ((3,1);(0,0);T_{2,3}^3\bigr )\);
 (I\(_{20})\)
 (I\(_{21})\)
 (I\(_{22})\)

\(V(4;0;3A_2)\);
 (I\(_{23})\)
 (I\(_{24})\)

\(V(4;0;A_{6})\);
 (II)
irreducible 2rigid families:
 (II\(_1)\)
 (II\(_2)\)

\(V(4;0;T_{3,4})\).
Proof
It is well known that if \(\deg \overline{C}\le 3\) then only smooth cubics are not rigid. All other families \(V(\mathbf{d};\mathbf{g}; \mathbf{S})\) of curves of degree \(\le 3\) are listed in \((\mathrm{I}_1)\)–\((\mathrm{I}_9)\). Therefore we assume below that \(\deg \overline{C}=4\).
Again, it is well known that if \(\overline{C}\) consists of four lines or two lines and a quadric, then \(\overline{C}\) is not rigid if and only if \(\overline{C}\) consists of four lines having a common point or \(\overline{C}\) consists of a quadric Q and two lines in general position with respect to Q. The rigid families in the case when \(\overline{C}\) consists of four lines or two lines and a quadric is listed in \((\mathrm{I}_{10})\)–\((\mathrm{I}_{14})\).
Consider the case when \(\overline{C}\) consists of two irreducible components: either \(\overline{C}=Q_0\cup Q_1\), where \(Q_0\) and \(Q_1\) are smooth quadrics, or \(\overline{C}=C\cup L\), where C is a cubic and L is a line.
Consider the case when \(\overline{C}_1=Q_0\cup Q_1\in V((2,2);(0,0);\mathbf{S})\). We have \(\mathbf{S}=m_1A_1+m_3A_3+m_5A_5+m_7A_7\), where \(m_1+2m_3+3m_5+4m_7=4\). The singularity type \(\mathbf{S}\) consists of \(k=m_1+m_3+m_5+m_7\), \(1\le k\le 4\), singular points and if \(m_1=4\), that is, \(\mathbf{S}=4A_1\), then \(V((2,2);(0,0);4A_1)\) is not rigid by (1), since \(\dim V((2,2);(0,0);4A_1)=10\). Similarly, if \(m_1=2\), that is, \(\mathbf{S}=A_3+2A_1\), then Open image in new window is not rigid, since Open image in new window . So, we can assume that \(m_1\le 1\).
Note that if \(m_3=2\), then the degenerate element \(Q_{1}\) of the pencil \(Q_{\lambda }\) consists of two lines \(z_2=0\) and \(z_3=0\) [see (4)].
In the case \(m_1=m_5=1\) (case \((\mathrm{I}_{15}))\) the strong rigidity of the curve \(\overline{C}_1\) follows from the equality \(\overline{C}_{\lambda }=Q_0\cup Q_{\lambda }=h_{\lambda }(\overline{C}_1)\) for each \(\lambda \in \mathbb {C}^*\), where the automorphism \(h_{\lambda }\) acts as follows: \(h_{\lambda }(z_1\!:\!z_2\!:\!z_3)=(\lambda z_1\!:\! \lambda ^2 z_2\!:\!z_3)\); and in the case \(m_7=1\) (case \((\mathrm{I}_{16}))\) the strong rigidity of the curve \(\overline{C}_1\) follows from equality \(\overline{C}_{\lambda ^2}=Q_0 \cup Q_{\lambda ^2}=h_{\lambda }(\overline{C}_1)\) for each \(\lambda \in \mathbb {C}^*\).
Consider the case when \(\overline{C}=C\cup L\), where C is a cubic and L is a line. It is easy to see that \(\overline{C}\) is rigid only if C is a rational curve. Therefore we have two cases: C is a nodal cubic or C is a cuspidal cubic.
Let C be a nodal cubic. Then (case \((\mathrm{I}_8)\)) \(V(3;0;A_1)\) is strongly rigid and \(\dim V(3;0;A_1)=8\). Therefore, L must be a “very special line” with respect to C, that is, L is either the tangent line of one of two branches of the node of C or L is the tangent line of C at a flex point of C. In both two cases it is easy to see that \(\overline{C}\) is strictly rigid and we have two cases: \((\mathrm{I}_{17})\)–\((\mathrm{I}_{18})\).
If C is a cuspidal cubic, then L is a tangent line to C or it passes through the cusp of C, since \(V(3;0;A_2)\) is strongly rigid and \(\dim V(3;0;A_2)=7\). If L is a tangent line to C at its cusp, then we have case (\(\mathrm{I}_{19}\)) and if L is the tangent line at the flex point of C, then we have case \((\mathrm{I}_{20})\). It is easy to see that in both cases \(\overline{C}\) is strongly rigid.
Let L be not the tangent line at the cusp of C and it pass through the cusp of C. Then if \(z_2=0\) is an equation of L, \(z_1=0\) is the line tangent to C at its cusp, and the line given by the equation \(z_3=0\) is tangent to C at its nonsingular point p of the intersection \(C\cap L\), then again we can assume that C is given by parametrization (5) and there are two possibilities: either \(a=0\) or \(a\ne 0\). Denote by \(C_a\) a curve given by parametrization (5). It is easy to see that \(p\in C_0\cap L\) is the flex point of \(C_0\) if \(a=0\) and \(p\in C_a\cap L\) is not the flex point of \(C_a\) if \(a\ne 0\). Therefore there is not a linear transformation Open image in new window such that \(h(C_0\cup L)=C_a\cup L\). On the other hand, the linear transformation \(h:(z_1\!:\!z_2\!:\!z_3)\rightsquigarrow (z_1\!:\!a^{1}z_2\!:\!a^{3}z_3)\) sends \(C_a\cup L\) to \(C_1\cup L\). Therefore \(V((3,1);(0,0), T_{2,3}^2)\) (case \((\mathrm{II}_1)\)) is an irreducible 2rigid family.

\(m_2=3\) and \(m_i=0\) for \(i\ne 2\),

\(m_2=m_4=1\) and \(m_i=0\) for \(i\ne 2,4\),

\(m_6=1\) and \(m_i=0\) for \(i\ne 6\),

\(m_9=1\) and \(m_i=0\) for \(i\le 8\).
The strict rigidity of a curve \(\overline{C}\in V(4;0;3A_2)\) (case \((\mathrm{I}_{22})\)) is well known.
As in case \((\mathrm{II}_1)\), there are two possibilities: either \(a=0\) or \(a\ne 0\).
Denote by \(\overline{C}_a\in V(4;0;T_{3,4})\) a curve given by (11). It is easy to see that \(\overline{C}_0\) has the unique flex point, namely, \(p_1=(1\!:\!0\!:\!0)\), and one can check that the curve \(\overline{C}_1\) has two flex points, \(p_1\) and Open image in new window . Hence there is no projective transformation Open image in new window such that \(\overline{C}_0=h(\overline{C}_1)\).
Remark 2.2
3 Strictly rigid rational curves of degree \(\ge \) 5
Theorem 3.1
For each \(n\ge 2\) the family \(V_n=V(2n; 0;3T_{n,n+1}+\cdots )\) is strictly rigid. The nonessential part of singularities of \(C\in V_n\) consists of simple singularities.
To prove Theorem 3.1 we need in the following result.
Theorem 3.2
We prove Theorems 3.1 and 3.2 simultaneously.
Proof of Theorems 3.1 and 3.2
Theorem 3.1 in the case \(n=2\) and Theorem 3.2 in the case \(n\le 3\) are well known, and we prove the general case by induction on n.
Remark 3.3
For \(n\le n_0+1\), it follows from strong rigidity that if Open image in new window , \(i=1,2\), are two curves such that \(C_1\) and \(C_2\) have two common flex points, say \(p_1\) and \(p_2\), then their third flex points \(p_3=C_1\cap L_3\) and \(q_3=C_2\cap L_3\) should coincide, \(p_3=q_3\), and consequently \(C_1=C_2\).
To complete the proof of theorems, note that if \(\overline{C}=C \cup L_1\cup L_2\cup L_3\) belongs to Open image in new window for \(n\le n_0+1\), then \(\sigma (C)\in V_{n+1}\), where \(\sigma \) is a quadratic transformation of the plane with centers at vertices of the triangle \(L_1\cup L_2\cup L_3\). Therefore Theorem 3.1 in the case \(n=n_0+1\) follows from Theorem 3.2 and Remark 3.3. \(\square \)
The following theorem provides an infinite series of strictly rigid families parameterizing the unions of two rational curves.
Theorem 3.4
The families Open image in new window , where \(n_{(1,2)}=n_{(1)}=1\), are strictly rigid for \(n\ge 2\).
Proof
Lemma 3.5
Proof
If \(k=1\) then it is easy to see that \(P_1(t)=t+1\). Assume that for \(k<k_0\) lemma is true and prove it in the case \(k=k_0\).
4 Strictly rigid curves of positive genera
The following theorem states that the strongly rigid family \(V(4;0;3A_2)\) [see case \((\mathrm{I}_{22})\) in Theorem 2.1] is the first member in an infinite sequence of strictly rigid families.
Theorem 4.1
For any \(n\ge 2\), the family Open image in new window is nonempty and strictly rigid.
Proof
Lemma 4.2
There exists a unique curve in \({\mathbb {C}}^2\), given by (14) and having a singularity of type \(A_n\) in \(({\mathbb {C}}^*)^2\). It is irreducible and has genus \([n/2]1\).
Proof
Under assumption that C has no other singularities in \({\mathbb {C}}^2\), the genus value follows from the fact that \(\delta (T_{n,2n1})=(n1)^2\) and \(\delta (A_n)=[(n+1)/2]\).
The following theorem will be used in the proof of Theorem 5.3 (see Sect. 5.2).
Theorem 4.3
Proof
Consider a curve \(\overline{C}=C_1\cup C_2\cup C_3\cup C_4\in V_n\) and denote the components \(C_i\) of \(\overline{C}\) as follows: \(C_1=C\) and \(C_i=L_{i1}\) for \(i\ge 2\). The curves \(L_1, L_2, L_3\) are lines and C is a curve of degree \(n(n1)\). The curve C has 3n singular points of the singularity type \(T_{n1,n}\), since \(n_{(1,2)}=n_{(1,3)}=n_{(1,4)}=n\), and the set of all other singularities of C is \(\mathbf{S}_{F_n}\), that is, the singularity type of C is the same as the singularity type of the dual curve to the Fermat curve of degree n. Therefore, the dual curve \(\widehat{C}\) to C is nonsingular and \(\deg \widehat{C}=n\), and to prove theorem, it suffices to show that there are homogeneous coordinates in \({\mathbb {P}}^2\) such that in this coordinate system the curve \(\widehat{C}\) is the Fermat curve.
The curve \(\widehat{C}\) has 3n flex points given in local coordinates by the equation \(y=x^n\) and for \(i=1,2,3\) there are n lines \(L_{i,1},\dots , L_{i,n}\) from the pencil of lines passing through \(p_i\) such that they are tangent to \(\widehat{C}\) at its flex points.
It easily follows from (27) and (30) that the case, when the only one number either a or b, or c is equal to zero, is also impossible.
In the case when \(a=b=0\) and \(c\ne 0\) we have \(F(z_1,z_2,z_3)=z_1^n+(z_2\pm z_3)^n\). But, it is impossible since C is an irreducible curve.
The cases \(a=c=0\), \(b\ne 0\) and \(b=c=0\), \(a\ne 0\) are also impossible since in these cases we have, respectively, that \(F(z_1,z_2,z_3)=(z_1+bz_3)^n+ z_2^n\) since \(b^n=1\) or \(F(z_1,z_2,z_3)=(z_1+az_2)^n+ z_3^n\) since \(a^n=1\). As a result, we obtain that \(a=b=c=0\), that is \(F(z_1,z_2,z_3)=z_1^n+ z_2^n+z_3^n\). \(\square \)
5 krigid curves with \(k\ge 2\)
5.1 2rigid irreducible families of equisingular plane curves of degree \(\ge \) 5
An infinite series of irreducible 2rigid families is given in the following two theorems.
Theorem 5.1
The families Open image in new window are 2rigid and irreducible for \(n\ge 2\).
Proof
Theorem 5.2
The families Open image in new window are 2rigid and irreducible for \(n\ge 3\).
Proof
5.2 krigid families parameterizing curves with k connected components
Theorem 5.3
For each \(k\in {\mathbb {N}}\) there is a krigid family of equisingular plane curves consisting of k irreducible components.
Proof
Consider a curve \(\widetilde{C}=\bigcup _{i=1}^6C_i\in \overline{V}_n\). Denote the curve \(C_1\) by C and the curve \(C_i\) by \(L_{i1}\) for \(i\ge 2\). The curves \(L_1, \dots , L_5\) are lines and C is a curve of degree \(n(n1)\) and it is easy to see that \(\overline{C}=C\cup L_1\cup L_2\cup L_3\in V_n\), where \(V_n\) is the family of plane curves from Theorem 4.3. By Theorem 4.3, we can assume that C is the curve dual to the Fermat curve of degree n and \(L_i\) is given by the equation \(z_i=0\) for \(i=1,2,3\). Then it follows from the singularity type of the curve \(\widetilde{C}\) that \(L_4\) and \(L_5\) have, respectively, equations \(z_2+\varepsilon ^{m_1}z_3=0\) and \(z_2+\varepsilon ^{m_2}z_3=0\), where \(m_1\not \equiv m_2(\mathrm{mod}\, n)\) and \(\varepsilon \) is a primitive root of the equation \(x^n+1=0\).
Consider two curves \(\widetilde{C}_i=\overline{C}\cup L_{4,i}\cup L_{5,i}\in \overline{V}_n\), \(i=1,2\), where \(L_{4,i}\) is given by the equation \(z_2+\varepsilon ^{m_{1,i}}z_3=0\) and \(L_{5,i}\) is given by the equation \(z_2+\varepsilon ^{m_{2,i}}z_3=0\). It is easy to see that a projective transformation Open image in new window such that \(\widetilde{C}_2=h(\widetilde{C}_1)\) exists if and only if Open image in new window . Hence \(\overline{V}_n\) is a \((n1)/2\)rigid family of plane curves consisting of \((n1)/2=k\) irreducible components. \(\square \)
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