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Let $X$ be a hypersurface of degree $r$ in $\mathbb{P}^n$, and $Z\subset X$ be a closed subscheme of pure dim 1. Let $g(Z):=1-\chi(\mathcal{O}_Z)$ and $d(Z)$ be its degree. I'm wondering that is there any bound $g(Z)\leq F_{r,n}(d(Z))$ where $F_{r,n}(-)$ is a degree two polynomial?

When $Z$ is irreducible and reduced, I think this follows from some classical results of curves in $\mathbb{P}^n$. But what will happen when $Z$ is reducible, or even non-reduced?

The main example I consider is a degree five smooth hypersurface in $\mathbb{P}^4$.

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  • $\begingroup$ I do not quite understand this question. For $n\geq 4$, it is quite possible for such a hypersurface to contain, for every integer $g\gg 0$, a degree-$2$, nonreduced curve $Z$ whose reduced scheme is a line and whose nilradical is an invertible sheaf on that line of degree $-g-1$. Do you want to assume that $Z$ is reduced? $\endgroup$
    – Jason Starr
    Commented Jul 19, 2022 at 17:22
  • $\begingroup$ @JasonStarr Thanks! Does this example exist on quintic 3fold? The main example I am considering is quintic 3fold, and I find such inequality for all closed subschemes of dim 1(possibly non-reduced) will be really useful. But I did not find this in books or papers. Maybe it is well-known that one can get a bound of $g$ by $F(d)$ for all one-dimensional closed subschemes, using the results for integral curves? $\endgroup$
    – Kim
    Commented Jul 20, 2022 at 7:24
  • $\begingroup$ I said something wrong in my previous comment. There is a quadratic bound for all reduced curves. $\endgroup$
    – Jason Starr
    Commented Jul 20, 2022 at 10:04

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For a reduced curve $Z$, a general linear projection from $Z$ to $\mathbb{P}^2$ is a birational morphism to a plane curve $C$ of degree $d=d(Z)$. Thus, $$1-\chi(Z,\mathcal{O}_Z) \leq 1-\chi(C,\mathcal{O}_C) = d(d-2)/2.$$ In general, this is the best possible inequality, since some hypersurfaces of degree $r$ in $\mathbb{P}^n$ contain a $2$-plane, and thus contain plane curves $C$ of degree $d$.

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  • $\begingroup$ Thanks a lot! I am also wondering that is it possible to bound $g$ by $d$ of $C$, when $C$ is on a normal quintic surface in $\mathbb{P}^3$ (possibly singular)? $\endgroup$
    – Kim
    Commented Jul 21, 2022 at 6:10
  • $\begingroup$ The argument in my answer applies to all hypersurfaces (singular or not) of every degree in every projective space of dimension at least $3$. $\endgroup$
    – Jason Starr
    Commented Jul 22, 2022 at 10:43

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