A bound on the degree of schemes defined by quadratic equations

TL;DR

This paper establishes new bounds on the degree of complex projective schemes defined by quadratic equations, improving previous bounds under certain syzygy-related conditions, and provides classification results for cases of equality.

Contribution

It introduces a bound on the degree of schemes defined by quadrics based on codimension, weaker than existing properties, with improved asymptotic behavior and classification results.

Findings

Degree bound improves the classical $d \,\leq\, 2^c$ bound.

New bounds depend on properties $N_p$ and $N_{2,p}$.

Classification results for schemes achieving equality.

Abstract

We consider complex projective schemes $X\subset\Bbb{P}^{r}$ defined by quadratic equations and satisfying a technical hypothesis on the fibres of the rational map associated to the linear system of quadrics defining $X$. Our assumption is related to the syzygies of the defining equations and, in particular, it is weaker than properties $N_2$, $N_{2,2}$ and $K_2$. In this setting, we show that the degree, $d$, of $X\subset\Bbb{P}^{r}$ is bounded by a function of its codimension, $c$, whose asymptotic behaviour is given by ${2^c}/{\sqrt[4]{\pi c}}$, thus improving the obvious bound $d\leq 2^c$. More precisely, we get the bound $\binom{d}{2}\leq\binom{2c-1}{c-1}$. Furthermore, if $X$ satisfies property $N_p$ or $N_{2,p}$ we obtain the better bound $\binom{d+2-p}{2}\leq\binom{2c+3-2p}{c+1-p}$. Some classification results are also given when equality holds.