# Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?

Let $$k$$ be an algebraically closed field (I am mainly interested in $$k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$$). Consider a plane curve $$C \subset \mathbb{A}^2$$ of degree $$d$$ over the rational function field $$k(t)$$. Suppose that $$C$$ is absolutely irreducible, i.e., irreducible over $$\overline{k(t)}$$.

Am I right that there is only the finite number $$n$$ of elements $$t \in k$$ such that the reduction of $$C$$ to $$k$$ is reducible ? Is there an upper bound for $$n$$?

• Yes, $n$ is finite. No, there is no bound. If $a(t)$ is a polynomial with $n$ distinct roots, then $x^2+y^2=a(t)$ has degree $2$ and $n$ reducible fibers. Commented Feb 26, 2021 at 21:48
• Is there an upper bound depending on degrees of the coefficients of $C$? In your example this is $n$. Commented Feb 27, 2021 at 22:31
• The key fact you need is that being geometrically integral is a constructible property. This question: mathoverflow.net/questions/221921/… is the analogous question over $\mathbb{Q}$ and the references given there also answer it in your case. Commented Feb 28, 2021 at 17:51

The vector of coefficients of a curve of degree $$d$$ gives a point in $$\mathbb{P}^{N_d}, N_d = (d+1)(d+2)/2-1$$. Multiplication of equations defines a map $$\mathbb{P}^{N_r} \times \mathbb{P}^{N_{d-r}} \to \mathbb{P}^{N_d}$$. The union of the images of these maps for $$1 \le r \le d-1$$ is a proper closed subset $$X$$ of $$\mathbb{P}^{N_d}$$ consisting of the reducible equations.
A curve over $$k(t)$$ can be viewed as a map $$\mathbb{P}^{1} \to \mathbb{P}^{N_d}$$ over $$k$$. If its image is contained in $$X$$ then the curve is not absolutely irreducible. Otherwise it only meets $$X$$ in a finite number $$n$$ of points and $$n$$ can be bounded as a constant depending on $$d$$ times the maximum of the degrees in $$t$$ of the coefficients of the curve.
• Please clarify the last part: $n$ can be bounded as a constant depending on d times the maximum of the degrees in $t$ of the coefficients of the curve. Commented Mar 2, 2021 at 9:11