On the computation of the minimal resolution of smooth parametric varieties

In this paper we give some results on the computation of the minimal resolution of the ideal $I(V)$ of a smooth parametric variety $V\subset\mathbb{P}^n_k$ of dimension $m$ represented by homogeneous polynomials of the same degree $r$ and without base points. We show that the Castelnuovo-Mumford regularity of $V$ is $reg(V) = min\{d\geq m - \lfloor \frac{m}{r}\rfloor\vert H_V(d) = {dr+m\choose m} \} + 1$, where $H_V(d)$ is the Hilbert function of $V$ . If $V$ has maximal rank and the minimal degree of a generator of $I(V)$ is $\alpha\geq m-\lfloor \frac{m}{r}\rfloor$ then $reg(V)\leq\alpha + 1$. In this case the shifts of the free modules $F_i$ of a minimal free resolution of $I(V)$ are at most two and we show that the Betti numbers are determined by computing the linear part of the resolution. In particular, if $V$ is minimally resolved the $F_i$ have one shift, for all but one $i$. We show that if $(a_1, ..., a_q)\in k^q$ are the coefficients of the polynomials that represent $V$ there is an open subset $U$ of $\mathbb{A}^q_k$ such that, if $(a_1, ..., a_q)\in U$, $V$ is minimally resolved and that it is possible to check that $U$ is non-empty for fixed $m, n, r$ by computer.