The central theorem of this paper is a result on the degree where the higher non-zero homology module of the Koszul complex constructed with homogeneous polynomials over a field becomes non trivial. This result has a straightforward corollary on the complexity of the determination of the dimension of a projective variety. Let us state precisely our result about the Koszul complex. If $P_{1},\ldots ,P_{s}$ are homogeneous polynomials of $A=k[X_{0},\ldots ,X_{n}]$ and $d_{i}=\deg P_{i}$, the Koszul complex : $ \mathbb {K}:= 0\longrightarrow \wedge ^{s} A^{s} \stackrel {\mathrm {d} _{s}}{\longrightarrow } \wedge ^{s-1}A^{s} \stackrel {\mathrm {d} _{s-1}}{\longrightarrow } \cdots \stackrel {\mathrm {d} _{2}}{\longrightarrow } \wedge ^{1}A^{s} \stackrel {\mathrm {d} _{1}}{\longrightarrow } A\longrightarrow 0, $ $\mathrm {d} _{p}(e_{i_{1}} \wedge \cdots \wedge e_{i_{p}}):= \sum _{k=1}^{p}(-1)^{k+1} P_{i_{k}} e_{i_{1}} \wedge \cdots \wedge \widehat {e_{i_{k}}} \wedge \cdots \wedge e_{i_{p}} $ (with the notation $A^{s}=e_{1}A\oplus \cdots \oplus e_{s}A$) is graded if we put $\deg (Pe_{i_{1}}\wedge \cdots \wedge e_{i_{p}})=\deg P+ d_{i_{1}}+\cdots + d _{i_{p}}$, and the differentials are of degree zero. If we consider the degree $\nu $ part of this complex, we obtain a complex of $k$-vector spaces of finite dimensions that we shall note $\mathbb {K}_{\nu }$. It is well known that if the homogeneous ideal $I=(P_{1},\ldots ,P_{s})$ is different from $A$ (i.e. if $\deg P_{i}>0$ for all $i$), then $H_{p}(\mathbb {K} )=0$ for all $p>s-\mathrm {ht}(I)$ and $H_{p}(\mathbb {K} )\not = 0$ for $p=s-\mathrm {ht} (I)$, this is proved for instance in the book of Northcott [N, chap. 8, § 5, thm 6]. As $H_{p}(\mathbb {K})=\bigoplus _{\nu }H_{p}(\mathbb {K}_{\nu })$ we can conclude that $H_{p}(\mathbb {K}_{\nu })=0$ for all $\nu $ if $p>s-\mathrm {ht} (I)$ and $H_{p}(\mathbb {K}_{\nu } )\not = 0$ for some $\nu $ if $p=s-\mathrm {ht}(I)$. Our result is an effective version of this last result, namely Theorem : With the above notations and hypotheses, suppose for example that $ d _{1}\geq d_{2}\geq \cdots \geq d_{s}>0$, and let us put $r=\mathrm {ht} (I)$ and $\pi _{r}=d_{1}\cdots d_{r}$, $\sigma _{r}=d_{1}+\cdots +d_{r}-r$. Then :

1. $H_{p}(\mathbb {K}_{\nu } )=0$ for all $\nu $ if $p>s-r$ ;
2. if $r\leq n$, $H_{s-r}(\mathbb {K} ,\nu )\not = 0$ for all $\nu \geq \max \Bigl \{ \sigma _{r}+1,{{(n-r)\pi _{r}\sigma _{r}}\over {2\deg I}}\Bigr \} +r d _{r+1}+ d _{r+2} +\cdots + d _{s}; $
3. if $r=n+1$, there exists $\nu \leq d_{1}+\cdots +d_{s}-r$ such that $H_{s-r}(\nu )\not = 0$.

Let us remark that it is easy to determine if we are in the case $(c)$, as in this case $I_{\nu }=A_{\nu }$ for all $\nu >\sigma _{r}$ (and reciprocally). As $\deg I\geq 1$ we know that for $\nu =\max \{ \sigma _{r}+1,\frac 12(n-r) \pi _{r}\sigma _{r}\} +r\mathrm {d} _{r+1}+\mathrm {d} _{r+2} +\cdots +\mathrm {d} _{s}$, $H_{s-r}(\mathbb {K}_{\nu })=0$ if and only if $\mathrm {ht} (I)\not = r$. The determination of $H_{s-r}(\mathbb {K}_{\nu })$ is easy linear algebra over $k$, as the linear maps are defined by the formula above, and so the expression on the natural bases of the $k$-vector spaces $( \wedge ^{p}A^{s})_{\nu }$ is simple and the corresponding matrices are block matrices, each block is either zero or corresponds to the multiplication by one of the $P_{i}$'s in a degree $\leq \nu $. The idea for proving the theorem is to take $r$ general linear combinations of the generators and to examine how deep they are close to the ideal on the components of maximal dimension of the variety defined by $I$, and what they define outside. Our results were greatly inspired by the work of Amoroso [A]. The first remark is that $r$ linear combinations defines a regular sequence (this is due to Kronecker), and so, by a theorem of Macaulay, the associated ideal $I_{r}$ is unmixed of height $r$. The second fact is that $I_{r}$ is reduced (and even smooth) outside the support of $I$ ; this is consequence of a refined version of Bertini's theorem that can be found in the book of Jouanolou [J]. And the third property comes from the study of the blowing-up of the ideal $I$ and says that for all prime $\mathfrak P$ associated to $I$ of height $r$, the integral closures of the localizations of $I$ and $I_{r}$ at $\mathfrak P$ are the same (this is mostly contained in [Z-S, vol. 2, chap. VII, § 8] and [N-R]). From that point, the proof is going on this way :

1. By a ical lemma (proved e.g. in [N] in the way of proving that $H_{s-r}(\mathbb {K} )\not = 0$) as $I_{r}$ is given by a regular sequence, $H_{s-r}(\mathbb {K}_{\nu })$ is isomorphic to $H_{s}(\overline {\mathbb {K}}_{\nu +\mathrm {d} _{1}+\cdots +\mathrm {d} _{r}})=\{ P\in (A/I_{r})_{\nu -(\mathrm {d} _{r+1}+\cdots +\mathrm {d} _{s})} \mid \forall i,\ PP_{i}\in I_{r} \}$ (here $\overline {\mathbb {K}}$ denotes the Koszul complex constructed on the quotient ring $A/I_{r}$).
2. From the second result quoted above, there exists two pure dimensional ideals $I'$ and $J$ of height $r$ such that $I_{r}=I'\cap J$ with ${\rm Supp}(I')\subseteq {\rm Supp}(I)$, $J=\sqrt {J}$ and the primes associated to $J$ are not in the support of $I$ (it is also possible that $I_{r}=I'$ in which case the proof is simpler and essentially skips the next step).
3. The Hilbert function of $I_{r}$ is easily expressible in terms of $d_{1},\ldots ,d_{r}$. Using the upper bound of [C] for the Hilbert function of the ideal $J$ and comparing to the expression for $I_{r}$, we are able to prove that for $\nu \geq \max \{ \sigma _{r}+1, (n-r)\pi _{r}\sigma _{r}/(2\deg I)\}$, $(J/I_{r})_{\nu }\not = 0$.
4. From a result of Briançon and Skoda on the integral closure of ideals, that was generalized by Lipman and Teissier [L-T], and the third property of general linear combinations quoted before, we deduce that $JI^{r}\subseteq I_{r}$ and the theorem follows.