A remark on the genus of curves in $${\mathbf {P}}^4$$ P 4
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A remark on the genus of curves in P4 Vincenzo Di Gennaro1 Received: 6 May 2019 / Accepted: 18 September 2019 / Published online: 24 September 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract Let C be an irreducible, reduced, non-degenerate curve, of arithmetic genus g and degree d, in the projective space P4 over the complex field. Assume that C satisfies the following flag condition of type (s, t): C does not lie on any surface of degree < s, and on any hypersurface of degree < t. Improving previous results, in the present paper we exhibit a Castelnuovo– Halphen type bound for g, under the assumption s ≤ t 2 − t and d t. In the range t 2 − 2t + 3 ≤ s ≤ t 2 − t, d t, we are able to give some information on the extremal curves. They are arithmetically Cohen–Macaulay curves, and lie on a flag like S ⊂ F, where S is a surface of degree s, F a hypersurface of degree t, S is unique, and its general hyperplane section is a space extremal curve, not contained in any surface of degree < t. In the case d ≡ 0 (modulo s), they are exactly the complete intersections of a surface S as above, with a hypersurface. As a consequence of previous results, we get a bound for the speciality index of a curve satisfying a flag condition. Keywords Genus of a complex projective curve · Castelnuovo–Halphen Theory · Flag condition Mathematics Subject Classification Primary 14N15 · 14N05; Secondary 14H99
1 Introduction Let C be an irreducible, reduced, non-degenerate curve, of arithmetic genus g and degree d, in the projective space P4 over the complex field. Assume that C satisfies the following flag condition of type (s, t): C does not lie on any surface of degree < s, and on any hypersurface of degree < t. Under the assumption s > t 2 − t and d > max(12(s + 1)2 , s 3 ), in [2, Theorem] it is proven a sharp upper bound G(d, s, t) for g (for the definition of G(d, s, t), see Sect. 2, (ii), below). In the present paper, we prove that this bound G(d, s, t) applies also when s ≤ t 2 − t and d t. More precisely, we prove the following:
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Vincenzo Di Gennaro [email protected] Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy
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Theorem 1.1 Let C be an irreducible, reduced, non-degenerate curve, of arithmetic genus g and degree d, in the projective space P4 over the complex field. Assume C is not contained in any hypersurface of degree < t (t ≥ 3), and in any surface of degree < s (s ≥ 3). Define α, β, m and by dividing s − 1 = αt + β, 0 ≤ β < t, and d − 1 = ms + , 0 ≤ < s. Assume s ≤ t 2 − t and d > d0 , where ⎧ 4 ⎪ ⎨32t if s ≤ 2t − 3, 4 (1) d0 := 8st if s ≥ 2t − 3 and β < t − α − 2, ⎪ ⎩ 2 3 max(12(s + 1) , s ) if s ≥ 2t − 3 and β ≥ t − α − 2. One has: • If either s ≤ 2t − 3 or s ≥ 2t − 3 and β < t − α − 2 or s ≥ 2t − 3 and β > t − α − 2 and either s − − 1 < α + β + 2 − t or β(α + β + 2 − t) ≤ s − − 1 < (β + 1)(α + β + 2 − t), then g < G(d, s, t); • Otherwise g < G(d, s, t) + 4t 3 . The proof relies on the
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