Non-normal surfaces of degree 5 and 6 in P n

Abstract

The purpose of this paper is to describe the non-normal surfaces of degree 5 and 6 embedded in the complex projective space P", with n i> 4. The idea is to study the normalization of such a surface, and then to find how a non-normal surface can be obtained from its normalization.Let Yc Pn be a surface of degree d, linearly normal and not contained in any hyperplane of P~. If d ~< 4, the list of all such surfaces is known (cf. 1-23], [24]). If 5 ~< d ~< 8 and Y is smooth, the classification is also known (cf.[15], 1-16]). The list of all normal singular surfaces of degree 5 and 6 is given in [10] and [11]. The purpose of this paper is to describe the non-normal surfaces of degree 5 and 6 embedded in P~.

Normal Projective Varieties of Degree 5

“…Finally, in Section 4 we deduce from the proof of the above Theorem some results which extend those in [5] and [8] to any algebraically closed field k of char(k) ≥ 0 (see Prop. 2 and Thm.…”

“…(4) a cone over a smooth rational curve of degree five; (5) (S, H) is a Gorenstein Del Pezzo surface of degree five with K S ≃ −H;…”

“…If X is a smooth complex projective variety, then the classification is also known (see P. Ionescu [7]). Furthermore, the classification of all complex normal surfaces was obtained by E. Halanay in [5]. Finally, we also mention here that the classification of all complex surfaces of degree 5 in P N C which are not contained in any hyperplane of P N C is known too (see, for example, G. Castelnuovo [2] and [3], J.G.…”

Normal surfaces of degree 7 in ? n

“…Since q(Y) = 0, it follows that q(Y) = 0 so that Y is normal, having only rational singularities (the last statement follows from Theorem 4 in [10], since the minimal desingularization of Y is a rational surface). EXAMPLE 11.…”

On first order congruences of lines in ℙ4 with irreducible fundamental surface

“…From Theorem 4 of [25] and Lemma 3.3, we get that the only possibility is that F is the projection of a (possibly singular) Del Pezzo surface in P 5 .…”