Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set $$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]\leq D,\;H_{K(\xi)}(\xi)\leq B\}.$$ and $$N(X;D,B)=\#S(X;D,B),$$ where $H_{K(\xi)}(\cdot)$ is the common naive height of an algebraic point over the field $K(\xi)$.

I guess that we might have $$N(X;D,B)\ll_{n,K,D}\delta N(\mathbb P^d_K;D,B),$$ since for the case of $D=1$, which is the rational points' case, we can prove the result. But I don't know how to prove it.

If this question is too difficult, at least could we prove the complete intersection case? I can only prove the case of hypersurfaces.

Thank you very much.

PS. In this question, the definition of algebraic points of bounded height is different from the usual definition, which is $$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]=D,\;H_{K(\xi)}(\xi)\leq B\}.$$

Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set $$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]\leq D,\;H_{K(\xi)}(\xi)\leq B\}.$$ and $$N(X;D,B)=\#S(X;D,B),$$ where $H_{K(\xi)}(\cdot)$ is the common naive height of an algebraic point over the field $K(\xi)$.

I guess that for all $X$ with degree $\delta$ and dimension $d$, we might have $$N(X;D,B)\ll_{n,K,D}\delta N(\mathbb P^d_K;D,B),$$ since for the case of $D=1$, which is the rational points' case, we can prove the result. But I don't know how to prove it.

If this question is too difficult, at least could we prove the complete intersection case? I can only prove the case of hypersurfaces.

Thank you very much.

PS. In this question, the definition of algebraic points of bounded height is different from the usual definition, which is $$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]=D,\;H_{K(\xi)}(\xi)\leq B\}.$$

Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set $$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]\leq D,\;H_{K(\xi)}(\xi)\leq B\}.$$ and $$N(X;D,B)=\#S(X;D,B),$$ where $H_{K(\xi)}(\cdot)$ is the common naive height of an algebraic point over the field $K(\xi)$.

I guess that we might have $$N(X;D,B)\ll_{n,K,D}\delta N(\mathbb P^d_K;D,B),$$ since for the case of $D=1$, which is the rational points' case, we can prove the result. But I don't know how to prove it.

If this question is too difficult, at least could we prove the complete intersection case? I can only prove the case of hypersurfaces.

Thank you very much.

Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set $$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]\leq D,\;H_{K(\xi)}(\xi)\leq B\}.$$ and $$N(X;D,B)=\#S(X;D,B),$$ where $H_{K(\xi)}(\cdot)$ is the common naive height of an algebraic point over the field $K(\xi)$.

I guess that we might have $$N(X;D,B)\ll_{n,K,D}\delta N(\mathbb P^d_K;D,B),$$ since for the case of $D=1$, which is the rational points' case, we can prove the result. But I don't know how to prove it.

If this question is too difficult, at least could we prove the complete intersection case? I can only prove the case of hypersurfaces.

Thank you very much.

PS. In this question, the definition of algebraic points of bounded height is different from the usual definition, which is $$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]=D,\;H_{K(\xi)}(\xi)\leq B\}.$$