Fix integers r &gt;= 4 and i &gt;= 2 (for r = 4 assume i &gt;= 3). Assume that the rational number s defined by the equation ((i + 1)(2))s + (i + 1) = ((r + i) )(i)is( ) an integer. Fix an integer d &gt;= s. Divide d - 1 = ms + epsilon, 0 &lt;= epsilon &lt;= s - 1, and set G(r;d, i) := ((m)(2))s + m epsilon. As a number, 2 G(r; d, i) is nothing but the Castelnuovo's bound G(s + 1;d) for a curve of degree d in Ps+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 P-r, of degree d &gt;&gt; max{ r,i}, not contained in hypersurfaces of degree &lt;= i. We prove that the bound G(r; d, i) is sharp if and only if there exists an integral surface S subset of P-r of degree s, not contained in hypersurfaces of degree &lt;= i. Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree s in Ps+1 The existence of such a surface S is known for r &gt;= 5, and 2 &lt;= i &lt;= 3. It follows that, when r &gt;= 5, and i = 2 or i = 3, the bound G(r; d, i) is sharp, and the extremal curves are isomorphic projection in P-r of Castelnuovo's curves of degree d in Ps+1. We do not know whether the bound G(r; d, i) is sharp for i &gt; 3.

Di Gennaro, V. (2022). On the genus of projective curves not contained in hypersurfaces of given degree. RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO [10.1007/s12215-022-00844-6].

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

#### Abstract

Fix integers r >= 4 and i >= 2 (for r = 4 assume i >= 3). Assume that the rational number s defined by the equation ((i + 1)(2))s + (i + 1) = ((r + i) )(i)is( ) an integer. Fix an integer d >= s. Divide d - 1 = ms + epsilon, 0 <= epsilon <= s - 1, and set G(r;d, i) := ((m)(2))s + m epsilon. As a number, 2 G(r; d, i) is nothing but the Castelnuovo's bound G(s + 1;d) for a curve of degree d in Ps+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 P-r, of degree d >> max{ r,i}, not contained in hypersurfaces of degree <= i. We prove that the bound G(r; d, i) is sharp if and only if there exists an integral surface S subset of P-r of degree s, not contained in hypersurfaces of degree <= i. Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree s in Ps+1 The existence of such a surface S is known for r >= 5, and 2 <= i <= 3. It follows that, when r >= 5, and i = 2 or i = 3, the bound G(r; d, i) is sharp, and the extremal curves are isomorphic projection in P-r of Castelnuovo's curves of degree d in Ps+1. We do not know whether the bound G(r; d, i) is sharp for i > 3.
##### Scheda breve Scheda completa Scheda completa (DC)
2022
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03
English
Projective curve
Castelnuovo-Halphen theory
Projection of a rational normal scroll surface
Maximal rank
Di Gennaro, V. (2022). On the genus of projective curves not contained in hypersurfaces of given degree. RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO [10.1007/s12215-022-00844-6].
Di Gennaro, V
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2108/320157`
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