The Manin-Peyre's conjectures for an infinite family of projective hypersurfaces in higher dimension

For a projective variety containing infinitely many rational points, a
natural question is to count the number of such points of height less
than some bound $B$. The Manin-Peyre's conjectures predict, for Fano
varieties, an asymptotic formula for this number as $B$ goes to
$+\infty$ in terms of geometric invariants of the variety. We will
discuss in this talk the Manin-Peyre's conjectures in the case of the
equation $$x_1y_2y_3\cdots y_n+x_2y_1y_3\cdots
y_n+\cdots+x_ny_1y_2\cdots y_{n-1}=0$$ for every $n \ge 2$.