The cohomological dimension of a resultant variety in prime characteristic \(p\) (Q1357797)

Let \(k\) be a field and \(F_1(s,t),\dots, F_n(s,t)\) homogeneous polynomials of \(s\) and \(t\) with indeterminate coefficients \(X_1,\dots, X_l\). We consider the ideal \(I\) of \(A= k[X_1,\dots, X_k]\) obtained by eliminating \(s\) and \(t\) from \((F_1,\dots, F_n)\). If \(I\) is generated by \(m\) elements, then \(H_I^i(A)=0\) for \(i> m\). Thus the cohomological dimension of \(V(I)\) is a lower bound of the number of generators of \(I\). In this article, the author computes the cohomological dimension of \(V(I)\) by using the theory of \(F\)-modules when \(\text{char } k>2\), \(n=3\) and \(\deg F_1=\cdots= \deg F_3= 2\). The case of \(\text{char } k=0\) is studied by \textit{Z. Yan} in another article.