Resolution of the ideal sheaf of a generic union of conics in P3:I

Abstract

We work over an algebraically closed field K of characteristic zero. Let Y be the generic union of $r \geq 2$ skew conics in $P^3_K$, $I_Y$ its ideal sheaf and v the least integer such that $h^0(I_Y(v)) > 0$. We first establish a conjecture (concerning a maximal rank problem) which allows to compute, by a standard method, the minimal free resolution of $I_Y$ if $r\geq 5$ and $\displaystyle{\frac{v(v+2)(v+3)}{12v+2}<r < \frac{(v+1)(v+2)(v+3)}{12v+6}}$. At the second time, we give the first part of the proof of that conjecture.