# Upper bound on the number of generators of a local complete intersection curve in $\mathbb{P}^3$

Let $C$ be a local complete intersection curve in $\mathbb{P}^3$ (not irreducible or smooth) of degree $e$. Suppose $f_1, f_2$ (and $e_i=\deg(f_i)$) are two of the lowest degree generators of $I(C)$. Then:

1) Is it true that the minimal number of generators of $I(C)$ is less than or equal to $2+(e_1e_2-e)$? (Intuitively I would think this to be true by using the degree formula from intersection theory which says that $e_1e_2=\sum_im_i\deg(C_i)$ where $C_i$ ranges over the irreducible components of the curve defined by $f_1$ and $f_2$ and $m_i$ are their multiplicities)

2) Is there any known result/approach to compute the upper bound on the minimal number of generators of $I(C)$ in terms of its degree?

• @Kalinin: I am aware of this result. I was asking for a bit stronger result which is to say the number of generators of the ideal of a curves i.e., the radical of this ideal generated by these $4$ equations. Is there a bound on the number of generators of the radical of an ideal based on the number of generators of the original ideal? Thanks anyways for your answer. Nov 15, 2012 at 11:19