the ideal generated by general polynomials is radical Let $F_1,\dots,F_s$, $s<n$, be homogeneous polynomials in $n+1$ variables of degrees $d_1,\dots,d_s$. 
Let $X$ be the intersection of the $s$ hypersurfaces of $\mathbb{P}^n$ defined
by $F_1,\dots,F_s$. By the homogeneous Hilbert's nullstellensatz, we know that 
$I_X = \operatorname{rad}(F_1,\dots,F_s)$. However, when $F_1,\dots,F_s$ are general, we have the remarkable fact that $(F_1,\dots,F_s)$ is radical. Can anyone please provide a proof (or sketch of proof) of this fact?
Edit/Definition: Recall that the space of polynomials of degree $d$ in $n+1$ variables is parametrized by $\mathbb{P}^{M_n}, M_n:={n+d \choose d}-1$. Then requiring $F_1,\dots,F_s$ to be general for the statement to hold, is interpreted as "there exists a Zariski open set $U$ of $\mathbb{P}^{M_{n_1}} \times \cdots \times \mathbb{P}^{M_{n_s}}$, such that the statement is true for every $s$-tuple of polynomials inside $U$".
PS: Since the relation between an ideal and its radical is in general a complicated one, i find the statement of the question extremely interesting and useful and i don't know why finding a proof in the literature seems so hard.
 A: First, recall a basic version of Bertini's theorem:
Theorem (Bertini): Let $Y$ be any nonsingular, irreducible projective variety; then its intersection $Y \cap H$ with a general hyperplane $H$ will be nonsingular. 
Now, suppose that we have $s \le n$ homogeneous polynomials $F_1,\dots,F_s$ of degrees $d_1,\dots,d_s$ in $n+1$ variables. Iterative application of the above theorem on the irreducible components of $Z(F_1,\dots,F_i)$ gives:
Corollary: If $F_1,\dots,F_s$ are general, then $X=Z(F_1,\dots,F_s)$ will be nonsingular.
By arguments involving the Hilbert function of $R = S/I, \, I=(F_1,\dots,F_s)$, one can show that 
Proposition: If $F_1,\dots,F_s$ are general, then $F_1,\dots,F_s$ is a regular sequence.
Note that the above proposition implies that $\operatorname{height}(I)=s$.
Next, let $J$ be the Jacobian ideal of $F_1,\dots,F_s$, i.e. the ideal generated by all $s \times s$ minors of the Jacobian matrix $\left[\frac{\partial F_i}{\partial x_j} \right]$. Since $X$ is nonsingular, $J \not\subset \mathfrak{m}_x, \, \forall x \in X$. This means that in the quotient ring $R = S/I$, we must have 
\begin{align}
\operatorname{height}(\bar{J}) = \operatorname{height}(\bar{\mathfrak{m}})=\operatorname{height}(\mathfrak{m})- \operatorname{height}(I) = n+1 -s \ge 1,\end{align}
where $\mathfrak{m}$ is the irrelevant ideal.
Finally, the key to deducing that $I$ is reduced is Serre's criterion for primality, e.g. Theorem 18.15 from Eisenbud, CA, 2004; a quite non-trivial result. According to that Theorem, if $R$ is Cohen Macaulay (and it is in our case since the $F_i$ form a regular sequence), then $I$ is reduced if and only if $\operatorname{height}(\bar{J}) \ge 1$. 
