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I am surprised nobody mentioned the result of Lu and Miyaoka-Lu (Math. Res. Letter 2, 663-676 (1995)) which implies indeed that there are only finitely many smooth rational curves on a surface of general type, thus answering the question of the OP.
I am surprised nobody mentioned the result of Miyaoka-Lu (Math. Res. Letter 2, 663-676 (1995)) which implies indeed that there are only finitely many smooth rational curves on a surface of general type, thus answering the question of the OP.
I am surprised nobody mentioned the result of Lu and Miyaoka (Math. Res. Letter 2, 663-676 (1995)) which implies indeed that there are only finitely many smooth rational curves on a surface of general type, thus answering the question of the OP.
I am surprised nobody mentioned the result of Miyaoka-Lu (Math. Res. Letter 2, 663-676 (1995)) which implies indeed that there are only finitely many smooth rational curves on a surface of general type, thus answering the question of the OP.
