$N_6$ property for third Veronese embeddings

TL;DR

This paper proves that third Veronese embeddings satisfy property N_6 by establishing the vanishing of specific rational homology groups of matching complexes, confirming a conjecture and highlighting the optimality of the result.

Contribution

It proves the Ottaviani-Paoletti conjecture for third Veronese embeddings, showing they satisfy property N_6 and do not satisfy N_7 for certain cases.

Findings

Proves vanishing of certain rational homology groups of matching complexes.

Confirms third Veronese embeddings satisfy property N_6.

Establishes the optimality of the N_6 property for these embeddings.

Abstract

The rational homology groups of the matching complexes are closely related to the syzygies of the Veronese embeddings. In this paper we will prove the vanishing of certain rational homology groups of matching complexes, thus proving that the third Veronese embeddings satisfy the property $N_6$. This settles the Ottaviani-Paoletti conjecture for third Veronese embeddings. This result is optimal since $\nu_3(\PP^n)$ does not satisfy the property $N_7$ for $n\ge 2$ as shown by Ottaviani-Paoletti in \cite{OP}.