Let $X \subset \mathbb{P}^n$ be a projective scheme. Denote by $I_X$ the (saturated) ideal of $X$. Suppose the degree of $X$ is $d$. Does there exist a polynomial in $I_X$ of degree less than or equal to $d$?

Let $X \subset \mathbb{P}^n$ be any projective scheme. Denote by $I_X$ the (saturated) ideal of $X$. Suppose the degree of $X$ is $d$. Under what assumptions there exists a polynomial in $I_X$ of degree less than or equal to $d$?