On the genus of projective curves not contained in hypersurfaces of given degree

Abstract

AbstractFix integers $$r\ge 4$$ r ≥ 4 and $$i\ge 2$$ i ≥ 2 (for $$r=4$$ r = 4 assume $$i\ge 3$$ i ≥ 3 ). Assume that the rational number s defined by the equation $$\left( {\begin{array}{c}i+1\\ 2\end{array}}\right) s+(i+1)=\left( {\begin{array}{c}r+i\\ i\end{array}}\right)$$ i + 1 2 s + ( i + 1 ) = r + i i is an integer. Fix an integer $$d\ge s$$ d ≥ s . Divide $$d-1=ms+\epsilon$$ d - 1 = m s + ϵ , $$0\le \epsilon \le s-1$$ 0 ≤ ϵ ≤ s - 1 , and set $$G(r;d,i):=\left( {\begin{array}{c}m\\ 2\end{array}}\right) s+m\epsilon$$ G ( r ; d , i ) : = m 2 s + m ϵ . As a number, G(r; d, i) is nothing but the Castelnuovo’s bound $$G(s+1;d)$$ G ( s + 1 ; d ) for a curve of degree d in $${\mathbb {P}}^{s+1}$$ P s + 1 . In the present paper we prove that G(r; d, i) is also an upper bound for the genus of a reduced and irreducible complex projective curve in $${\mathbb {P}}^r$$ P r , of degree $$d\gg \max \{r,i\}$$ d ≫ max { r , i } , not contained in hypersurfaces of degree $$\le i$$ ≤ i . We prove that the bound G(r; d, i) is sharp if and only if there exists an integral surface $$S\subset {\mathbb {P}}^r$$ S ⊂ P r of degree s, not contained in hypersurfaces of degree $$\le i$$ ≤ i . Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree s in $${\mathbb {P}}^{s+1}$$ P s + 1 . The existence of such a surface S is known for $$r\ge 5$$ r ≥ 5 , and $$2\le i \le 3$$ 2 ≤ i ≤ 3 . It follows that, when $$r\ge 5$$ r ≥ 5 , and $$i=2$$ i = 2 or $$i=3$$ i = 3 , the bound G(r; d, i) is sharp, and the extremal curves are isomorphic projection in $${\mathbb {P}}^r$$ P r of Castelnuovo’s curves of degree d in $${\mathbb {P}}^{s+1}$$ P s + 1 . We do not know whether the bound G(r; d, i) is sharp for $$i>3$$ i > 3 .