Rational points on algebraic curves that change genus (Q1377592)

Let \(K\) be an algebraic function field in one variable over an algebraically closed field of positive characteristic \(p\), and let \(\overline K\) denote the algebraic closure of \(K\). An algebraic curve \(C/K\) is said to be non-conservative or genus-changing if its relative genus \(g_K\) is different from the absolute genus \(\overline g:=g_{\overline K}\). \textit{P. Samuel} [``Lectures on old and new results on algebraic curves'', Tata Inst. Fund. Res. (Bombay 1966; Zbl 0165.24102)] proved that a genus-changing algebraic curve \(C\) of absolute genus \(\overline g\geq 2\) has only finitely many \(K\)-rational points. \textit{J. P. Voloch} [Bull. Soc. Math. Fr. 119, No. 1, 121-126 (1991; Zbl 0735.14018)] established the same finiteness result for algebraic curves of absolute genus \(0\) or \(1\), when the constant field of the function field \(K\) is finite. In this paper the author considers genus-changing algebraic curves of absolute genus \(0\) for general \(K\), and asks if they have only finitely many \(K\)-rational points. The main result of this paper is formulated as follows: Theorem: Let \(K\) be a function field in one variable over an algebraically closed field of positive characteristic \(p\). Then every non-conservative algebraic curve \(C\) over \(K\) has finitely many \(K\)-rational points, that is, the set \(C(K)\) of \(K\)-rational points is finite. The result is first established for an algebraic curve \(C\) of the form \(y^p=r(x)\) that admits genus change. From this result, the author deduces the finiteness for \(C(K)\) for every genus changing algebraic curve \(C\) over \(K\).