There are many interesting sequences of polynomials which contain
polynomials of arbitrarily high degree, for example classical
orthogonal polynomials. Most of them arise as characteristic polynomials
of some sequences of operators, or as polynomial solutions
of some differential equations.
What are some natural specific
sequences of plane (affine or projective) algebraic
curves which contain curves of arbitrarily high degree and genus?
One such example is Fermat's curves $x^n+y^n=1$. Lissajous
(a.k.a. Chebyshev)
curves are of arbitrary degree but they have zero genus. Sequences of hyperelliptic curves occur in the theory of integrable systems. What else?
I looked to the Catalog of Plane curves by D. Lawrence (Dover, 2014)
and to the book of Brieskorn and Knörrer, Plane algebraic curves,
and found only Lissajous curves, epitrochoids and
hypotrochoids (all of genus zero) as examples of arbitrarily high degree.
I understand that many examples can be constructed. But I am asking on some naturally occuring sequences, whatever it can mean. Of some historical significance or appearing in applications.
EDIT. Thanks to all who answered or commented. I am not marking this question as "answered" for a while, hoping for more examples. Of course, <a href="https://en.wikipedia.org/wiki/Classical_modular_curve>classical modular curves belong here, thanks to Felipe Voloch.
Let me mention my motivation for this question. For some time I am studying what can be called "Lamé modular curves" (surprisingly, there is no established name for them). Lamé functions
are solutions of Lamé's differential equation whose squares are polynomials. Existence of
such a solution imposes a polynomial equation connecting the modulus of the
torus $J$ and an "accessory parameter". These polynomials define a family of plane affine algebraic curves
which contains curves of arbitrary degree and genus, and their coefficients are integers.
Best Answer
How about the affine plane curves $\Phi_n(c,t)=0$ that classify $(c,t)$ such that $t$ is a point of exact period $n$ under iteration of the quadratic map $f_c(X)=X^2+c$? These are often called dynatomic curves and have been much studied in recent years, especially since describing their rational points is related to the dynamnical uniform boundedness conjecture. These curves are irreducible (Bousch) and there is a nice formula for their genus (Morton) showing the genus goes to infinity. There is even some work (Poonen, Doyle, ...) showing that the gonality also grows. For the basic construction, you can see for example Sections 4.1 and 4.2 of my book The Arithmetic of Dynamical Systems. More generally, people study the dynatomic curves for $X^d+c$.
(I've cheated a little, one needs to include a few extra points on the curve where the point $t$ has "formal period $n$," but actual period smaller than $n$. This is Milnor's terminology.)