Curves of high genus in projective space

• Main Author: Zompatori, Marina
• Language: en_US
• Published: Boston University 2019
• Subjects: Algebra
• Online Access: https://hdl.handle.net/2144/33610

Thesis (Ph.D.)--Boston University === PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at [e-mail]. Thank you. === Algebraic curves C in the projective space P^n 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 P^n and to study the geometry of such curves. By Castelnuovo's Theorem, the maximum value of g, if deg(C) = d, in P^n is known and is denoted by π(d, n). If g = π(d, n), C lies on a surface S ⊂ P^n such that deg(S) = n - 1. To study other curves C with g < π(d, n), Eisenbud and Harris arranged the possible values of g into intervals πα+1(d, n) < g < πα(d, n) , with α E Z+ and π0(d, n) := π(d, n), where πα (d, n) = maxc⊂s{g(C)} for any normal surface S with deg(S) = n + α - 1. In general, for C ⊂ P^n such that g > πα (d, n), they proved that if n ≥ 8 and d ≥ 2^n+1, then C lies on a surface S ⊂ P^n with deg(S) ≤ n - 2 + α. They also conjectured that the same result should hold for curves of any degree, provided that g > πα(d, n) and proved this conjecture for a = 1. We will focus on the case a = 2. In this case, by the Eisenbud-Harris conjecture, C should lie on a surface of degree n in P^n. We verify this in several special cases for C ⊂ F. 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 P^5. The intersection of such hypersurfaces is a surface S ⊂ P^5 with deg(S) < 7. By analyzing the maximum value of the genus of C for C ⊂ S ⊂ P^n and deg (S) = (n + 1) or (n + 2), we see that the curves C ⊂ P^5 we are analyzing cannot lie on a surface S with deg(S) = 6, 7. === 2031-01-01