Let $X$ be a very general smooth hypersurface of degree $d (\ge 5)$ in $\mathbb{P}^3$ and $Y$ be another smooth hypersurface of degree $d^{\prime}$, where $3 \le d^{\prime} \le (d-1)$ such that $X \cap Y$ is irreducible. Is there any effective bound on the number of node on $X \cap Y$? Of course, there is a bound given by the gennus of a smooth complete intersection curve of degree $dd^{\prime}$. I am looking for more effective bound.
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$\begingroup$ Just to clarify, is it assumed that $X \cap Y$ has at most nodal singularities? $\endgroup$– cgodfreyCommented Apr 11, 2019 at 17:27
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$\begingroup$ Yes. I hope if we allow other singularities then number of singular points will be less. $\endgroup$– user130022Commented Apr 12, 2019 at 6:05
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