Algebraic curves C in the projective space Pn are characterized by their degree d and genus g. We would like to know what g are possible for a curve of degree d in Pn and to study the geometry of such curves. By Castelnuovo's Theorem, the maximum value of g, if deg(C) = d, in Pn is known and is denoted by pi(d, n). If g = pi(d, n), C lies on a surface S ⊂ Pn such that deg(S) = n - 1. To study other curves C with g pi( d, n), Eisenbud and Harris arranged the possible values of g into intervals pialpha+1 (d, n) g pi alpha(d, n), with alpha ∈ Z+ and pi0(d, n) := pi( d, n), where pialpha(d, n) = maxC⊂ S{lcub}g(C){rcub} for any normal surface S with deg(S) = n + alpha - 1. In general, for C ⊂ Pn such that g > pialpha(d, n), they proved that if n ≥ 8 and d ≥ 2n+1, then C lies on a surface S ⊂ Pn with deg(S) ≤ n - 2 + alpha. They also conjectured that the same result should hold for curves of any degree, provided that g > pialpha(d, n) and proved this conjecture for alpha = 1. We will focus on the case alpha = 2. In this case, by the Eisenbud-Harris conjecture, C should lie on a surface of degree n in Pn . We verify this in several special cases for C ⊂ P5 . To do so, we study the systems cut out by quadric and cubic hypersurfaces on C and prove that C must lie on at least three or four quadrics in P5 . The intersection of such hypersurfaces is a surface S ⊂ P5 with deg(S) ≤ 7. By analyzing the maximum value of the genus of C for C ⊂ S ⊂ Pn and deg(S) = (n + 1) or ( n + 2), we see that the curves C ⊂ P5 we are analyzing cannot lie on a surface S with deg( S) = 6, 7.
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