Exercise $2$ from chapter $5$ of Eisenbud's Geometry of Syzygies book I am trying to solve exercise $2$ from chapter $5$ of Eisenbud's The Geometry of Syzygies book.The problem is as follows: 

Let $X$ be the union of two disjoint lines in $\mathbb P^3$, or a conic   contained in a plane in $\mathbb P^3$. Then $2=\mathrm{reg}(I_X) > \deg(X)-\mathrm{codim}(X)+1$.

 A: Let $f \in S=k[x_1,x_2,x_3,x_4]$ be a quadratic form. Then we have an isomorphism $(f) \cong S(-2)$ of graded $S$-modules, and so 
the regularity of $(f)$ is $2$ (irrespectively of whether the corresponding conic lies in a plane or not). 
Now let $\ell_1,\ell_2$ be two disjoint lines in $\mathbb{P}^3$, and so the ideal
of their union is $I_{\ell_1} \cap I_{\ell_2}$. Now $I_{\ell_1} = (p_1,p_2)$, where $p_1,p_2$ are linear forms, and similarly $I_{\ell_2} = (q_1,q_2)$. Thinking of the two lines as $2$-dimensional subspaces in $k^4$, since they are disjoint, their sum must be $k^4$. This implies that their orthogonal complements are disjoint. This in turn implies that the linear forms $p_1,p_2,q_1,q_2$ are linearly independent and so by a change of coordinates we can assume that 
$I_{\ell_1} = (x_1,x_2)$ and $I_{\ell_2} = (x_3,x_4)$, where $x_1,x_2,x_3,x_4$ are the indeterminates of the underlying polynomial ring. But then $I_{\ell_1} \cap I_{\ell_2} = I_{\ell_1} I_{\ell_2}$. Now, the regularity of the product of $d$ linear forms is $d$, by a result of Conca and Herzog, 2003. Consequently, the regularity of $I_{\ell_1} \cap I_{\ell_2}$ is $2$.
Regarding the inequality $\mathrm{reg}(I_X) > \deg(X)-\mathrm{codim}(X)+1$, note that the degree and the codimension of the two lines are each equal to $2$, while for the conic in a plane its degree is zero and its codimension is $1$.
