The monodromy problem for hyperelliptic curves

Abstract

We study the Dynkin diagrams associated to the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case $y^4+g(x)$, which is a generalization of the results of Christopher and Mardešić about the monodromy problem for hyperelliptic curves. Moreover, we solve the monodromy problem for direct sums of fourth degree polynomials.

Introduction

The interest on polynomial foliations in ${\mathbb{C}}^2$ arises as an approach to the Hilbert Sixteen Problem [16], [21]. These foliations are given by 1-forms $\omega =P(x,y)dy-Q(x,y)dx$, where P and Q are polynomials. The classical notation for the foliation associated to the form ω is $\mathcal{F}(\omega )$. In this context, a point $p\in {\mathbb{C}}^2$ is a singularity of $\mathcal{F}(\omega )$ if $P(p)=Q(p)=0$. We say that the singularity p is a center singularity if there is a local chart such that p is mapped to $0\in {\mathbb{C}}^2$, and a Morse function $f:({\mathbb{C}}^2,0)\to \mathbb{C}$ with fibers tangent to the leaves of $\mathcal{F}(\omega )$. The degree of a foliation $\mathcal{F}(\omega )$ is the greatest degree of the polynomials P and Q, and the space of foliations of degree d is denoted by $\mathcal{F}(d)$. The closure of the set of foliations in $\mathcal{F}(d)$ with at least one center is denoted by $\mathcal{M}(d)$.

It is known that $\mathcal{M}(d)$ is an algebraic subset of $\mathcal{F}(d)$ (e.g. [19, §6.1] [16]). The problem of describing its irreducible components is formulated by Lins Neto [20]. In [12], Y. Ilyashenko proves that the space of Hamiltonian foliation $\mathcal{F}(df)$, where f is a polynomial of degree $d+1$, is an irreducible component of $\mathcal{M}(d)$. In [17], H. Movasati considers the logarithmic foliations $\mathcal{F}\left({\sum }_{i=1}^s{\lambda }_i\frac{d{f}_i}{f_i}\right)$, with $f_i\in \mathbb{C}{[x,y]}_{\le d_i}$ and ${\lambda }_i\in {\mathbb{C}}^{⁎}$. He proves that the set of logarithmic foliations form an irreducible component of $\mathcal{M}(d)$, where $d={\sum }_{i=1}^s{d}_i-1$. Moreover, in [23], Y. Zare works with pullback foliations $\mathcal{F}(P^{⁎}\omega )$, where $P=(R,S):{\mathbb{C}}^2\to {\mathbb{C}}^2$ is a generic morphism with $R,S\in \mathbb{C}{[x,y]}_{\le d_1}$, and ω is a 1-form of degree $d_2$. Zare shows that they form an irreducible component of $\mathcal{M}(d_1(d_2+1)-1)$.

The main idea in the proofs of these assertions is to choose a particular polynomial f and consider deformations $df+\epsilon {\omega }_1$ in $\mathcal{M}(d)$. Then, it is necessary to study the vanishing of the abelian integrals ${\int }_{\delta }{\omega }_1$, where δ is a homological 1-cycle in a regular fiber of f. This integral is zero on the vanishing cycle associated to the center singularity. If the monodromy action on this cycle generates the whole vector space $H_1(f^{-1}(b),\mathbb{Q})$ for a regular value b, then the deformation is relatively exact to df.

The condition that the vanishing of the integral ${\int }_{\delta }{\omega }_1$ implies that ${\omega }_1$ is relatively exact to df, is known as the (*)-property (It was introduced by J.P. Françoise in [9]). The results of L. Gavrilov in [10], show that if we provided that the integral is zero over any cycle in a regular fiber, then ${\omega }_1$ is relatively exact. Therefore, if the subspace generated by monodromy action on the vanishing cycle δ is the whole space $H_1(f^{-1}(b),\mathbb{Q})$, then the $(⁎)-property$ is satisfied. This gives rise to the next natural problem, which is summarized by C. Christopher and P. Mardešić in [4] as follows.

Monodromy problem. Under which conditions on f is the $\mathbb{Q}$-subspace of $H_1(f^{-1}(b),\mathbb{Q})$ generated by the images of a vanishing cycle of a Morse point under monodromy equal to the whole of $H_1(f^{-1}(b),\mathbb{Q})$?

Furthermore, they show a characterization of the vanishing cycles associated to a Morse point in hyperelliptic curves given by $y^2+g(x)$, depending on whether g is decomposable (Theorem 4.7). This case is closely related with the 0-dimensional monodromy problem; by using the definition of Abelian integrals of dimension zero in [11]. For example, if we think in the Dynkin diagram associated with $y^2+g(x)$ and the one associated with $g(x)$, then we see that they coincide. However, the Dynkin diagram for $y^3+g(x)$ is a bit more complicated. Moreover, in the case $y^4+g(x)$ there is always a pullback associated to $y\to y^2$, thus the Dynkin diagram is expected to reflect this fact. For these two cases, we prove the following two theorems.

Theorem 1.1

Let g be a polynomial with real critical points, and degree d such that $4\nmid d$ and $d\le 100$. Consider the polynomial $f(x,y)=y^4+g(x)$, and let $\delta (t)$ be an associated vanishing cycle at a Morse point; then one of the following assertions holds.

• The monodromy of $\delta (t)$ generates the homology $H_1(f^{-1}(t),\mathbb{Q})$.

• The polynomial g is decomposable (i.e., $g=g_2\circ g_1$), and ${\pi }_{⁎}\delta (t)$ is homotopic to zero in $\{y^4+g_2(z)=t\}$, where $\pi (x,y)=(g_1(x),y)=(z,y)$. Or, the cycle ${\pi }_{⁎}\delta (t)$ is homotopic to zero in $\{z^2+g(x)=t\}$, where $\pi (x,y)=(x,y^2)=(x,z)$.

Theorem 1.2

Let g be a polynomial with real critical points, and degree d such that, $3\nmid d$ and $d\le 100$. Consider the polynomial $f(x,y)=y^3+g(x)$, and let $\delta (t)$ be an associated vanishing cycle at a Morse point; then one of the following assertions holds.

• The monodromy of $\delta (t)$ generates the homology $H_1(f^{-1}(t),\mathbb{Q})$.

• The polynomial g is decomposable (i.e., $f=g_2\circ g_1$), and ${\pi }_{⁎}\delta (t)$ is homotopic to zero in $\{y^3+g_2(z)=t\}$, where $\pi (x,y)=(g_1(x),y)=(z,y)$.

Some parts in the proof are done numerically using computer, thus we have the restriction $d\le 100$ in the degree of the polynomial g.

The monodromy problem for polynomials of degree 4, on the other hand, is very interesting, because the classification of the irreducible components of $\mathcal{M}(3)$ is still an open problem. In fact, the only case which has a complete classification is $\mathcal{M}(2)$ (see [8][3, p. 601]). For polynomials $f(x,y)=h(y)+g(x)$ where $\text{deg}(h)=\text{deg}(g)=4$, we determine in the Theorem 5.4, a relation between the subspaces of $H_1(f^{-1}(b),\mathbb{Q})$ generated by the monodromy action on the vanishing cycles, and the property of f being decomposable. In oder to do that, we provide an explicit description of the space of parameters of the polynomials $h(y)+g(x)$ which satisfies some conditions in the critical values.

Organization. In section 2, we provide some definitions in Picard-Lefschetz theory and describe the Dynkin diagram for direct sum of polynomials in two variable. In section 3, we analyze the particular case, in which there is only one critical value. For this case, we compute the vector space generated by the monodromy action on the vanishing cycles. In section 4, we prove Theorem 1.1, Theorem 1.2. Finally, in Section 5, we solve the monodromy problem for polynomials $h(x)+g(y)$ with $\text{deg}(h)=\text{deg}(g)=4$; in this case there is another pullback to be considered, associated with the map $(x,y)\to (xy,x+y)$. For this reason, we do not know a geometrical characterization of some of vanishing cycles which do not generate the whole $H_1(f^{-1}(b),\mathbb{Q})$.

Acknowledgments. I thank Hossein Movasati for suggesting to study the action of monodromy and for introducing me to the center problem. I thank Lubomir Gavrilov for hosting me at the University of Toulouse during a short visit and for his helpful discussions. I thank the reviewer for pointing out that the examples in this paper are generally non-hyperelliptic curves. For example, the generic curve of genus 3, ${\mathrm{\Gamma }}_t=\{y^4+g(x)=t\}$ where $\mathrm{deg}(g)=4$, is non-hyperelliptic curve. Namely, the canonical map

$${\mathrm{\Gamma }}_t\to {\mathbb{P}}^2[x:y:1]\to [x\frac{dx}{y^3}:y\frac{dx}{y^3}:\frac{dx}{y^3}]=[x:y:1]$$

is an embedding. Despite this, the title of the article refers to the case of hyperelliptic curves, since we were inspired in [4].