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I want to ask if there is a known classification of possible singularities of curves on a general (or very general) surface in $\mathbb{P}^3$.

It was shown in Proposition 3 of "Subvarieties of general hypersurfaces in projective space" by Geng Xu that every hyperplane section of a general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has at most either (1) 3 nodes, (2) 1 node and one cusp or (3) a tacnode.

Also by a dimension count, on a general surface of degree $d \geq 4$, there are curves in the linear system $\left|\mathcal{O}_S(n)\right|$ with at most $\dim \left|\mathcal{O}_S(n)\right|$ nodes.

I wonder if there is a similar result for all complete intersections other than hyperplane sections and singularities other than nodes?

Thanks in advance!

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  • $\begingroup$ Note that Xu's result applies to all curves on a general surface, not only to hyperplane sections, giving a (very rough) lower bound on the geometric genus. I don't know if this has been improved. $\endgroup$
    – abx
    Commented Nov 14, 2023 at 8:03

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