Let \(\mathbb {P}^1=\mathbb {P}^1_{k}\) with \(k\) an algebraically closed field and let \(\mathcal{Q}=\mathbb {P}^1\times \mathbb {P}^1\) be a smooth quadric. Let \(S=k[u,u',v,v']\) be the bigraded ring, \(X\) be a zero-dimensional scheme and \(I=I(X)\) its saturated bigraded ideal in \(S\). A zero-dimensional scheme \(X\subseteq \mathcal{Q}\) is said scheme-theoretically generated by \(r\) forms \(f_1,\dots,f_r\in S\) with \(\deg f_i=(a_i,b_i)\) if there exists a sheaves surjection \(\bigoplus_{i=1}^r \mathcal{O}_{\mathcal{Q}}(a_i,b_i)\overset{\phi}\rightarrow \mathcal{I}_X\rightarrow 0\) with \(\phi=(f_1,\dots,f_r)\). If \(r=2\), \(X\) is called scheme-theoretically complete intersection.
Since in general in \(\mathbb {P}^1\times \mathbb {P}^1\) a bigraded ideal \(I(X)\) generated by a regular sequence is not saturated, it is interesting to study zero-dimensional schemes \(X\) defined by ideals that are the saturation of bigraded ideals generated by regular sequence.
In this paper, the authors study the case \(r=2.\)
In particular, Section 3 is devoted to the minimal case, i.e., when \(X\), arising by the ideal \((f,g),\) with \(\deg f=(a,b)\) and \(\deg g=(c,d)\), is contained in two curves of type \((a+c,0)\) and \((0,b+d).\) They found that \(X\) is the union of two \(0\)-grids and describe a minimal free bigraded resolution, showing that \(X\) has only four minimal generators (in the minimal case).
Section 4 is devoted to the general case, i.e., the curves \(C\) and \(D\) are two general curves of bidegree \((a,b)\geq (1,1)\) and \((c,d)\geq (1,1)\), respectively. They compute the Hilbert function of \(X\) and prove that the saturated ideal of \(X\) can be obtained by saturating the row ideals or the column ideals.