Abstract Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field k . We show that if either $X_4$ contains linear subspace $\Lambda $ $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ or has double points along 3$ , smooth -rational point and is otherwise general, then unirational This improves previous results by A. ...

Let (X, L) be a polarized variety over number field K. We suppose that L is an hermitian line bundle. M non compact Riemann Surface and $$U\subset M$$ relatively open set. $$\varphi :M\rightarrow X(\mathbf{C})$$ holomorphic map. For every positive real T, let $$A_U(T)$$ the cardinality of set $$z\in U$$ such (z)\in X(K)$$ $$h_L(\varphi (z))\le T$$ . After revisitation proof sub exponential boun...

Given an integer γ≥2 and odd prime power q we show that for every large genus g there exists a non-singular curve C defined over