Annali dell’Università di Ferrara

, Volume 50, Issue 1, pp 151–165

# A result on resolutions of Veronese embeddings

• Elena Rubei
Article

## Abstract

This paper deals with syzygies of the ideals of the Veronese embeddings. By Green’s Theorem we know thatO P n (d) satisfies Green-Lazarsfeld’s PropertyN pd≥p, ∀n. By Ottaviani-Paoletti’s theorem ifn≥2, d≥3 and 3d−2≤p thenO P n (d) does not satisfy PropertyN p. The casesn≥3, d≥3, d<p<3d−2 are still open (exceptn=d=3). Here we deal with one of these cases, namely we prove thatO P n (3) satisfies PropertyN 4n. Besides we prove thatO P n (d) satisfiesN pn≥p iffO P n (d) satisfiesN p.

## Key words

syzyzgies Veronese embedding

14M25 13D02

## Sunto

L’argomento di questo articolo sono le sizigie degli ideali delle varietà di Veronese. Per il teorema di Green sappiamo cheO P n (d) soddisfa la proprietàN p di Green-Lazarsfeld ∀d≥p, ∀n. Per il teorema di Ottaviani-Paoletti sen≥2, d≥3 and 3d−2≤p alloraO P n (d) non soddisfa la ProprietàN p. I casin≥3, d≥3, d<p<3d−2 sono ancora aperti (eccetton=d=3). Qui consideriamo uno di tali casi, precisamente proviamo cheO P n (3) soddisfa la ProprietàN 4n. Inoltre proviamo cheO P n (d) soddisfaN pn≥p se e solo seO P p (d) satisfiesN p.

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