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Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$ such that the restriction of this linear system to $X$ yields $\Lambda$, plus some fixed components.

Is it true that the degree of $\Lambda'$ is bounded if the degree of $\Lambda$ is bounded? (the degree is here the degree in $\mathbb{P}^n$).

Comment: Denoting by $D\in \mathrm{Pic}(X)$ the divisor of $\Lambda$ and by $H\in \mathrm{Pic}(X)$ the very ample divisor given by the restriction of an hyperplane, we are looking for an integer $d$ such that $dH-\Lambda$ is effective. The question corresponds to ask if $d$ is bounded if the degree of $\Lambda$ is.

EDIT: The bound asked here can depend on $X$.

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$ such that the restriction of this linear system to $X$ yields $\Lambda$, plus some fixed components.

Is it true that the degree of $\Lambda'$ is bounded if the degree of $\Lambda$ is bounded? (the degree is here the degree in $\mathbb{P}^n$).

Comment: Denoting by $D\in \mathrm{Pic}(X)$ the divisor of $\Lambda$ and by $H\in \mathrm{Pic}(X)$ the very ample divisor given by the restriction of an hyperplane, we are looking for an integer $d$ such that $dH-D$ is effective. The question corresponds to ask if $d$ is bounded if the degree of $D$ is.

EDIT: The bound asked here can depend on $X$.

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$ such that the restriction of this linear system to $X$ yields $\Lambda$, plus some fixed components.

Is it true that the degree of $\Lambda'$ is bounded if the degree of $\Lambda$ is bounded? (the degree is here the degree in $\mathbb{P}^n$).

Comment: Denoting by $D\in \mathrm{Pic}(X)$ the divisor of $\Lambda$ and by $H\in \mathrm{Pic}(X)$ the very ample divisor given by the restriction of an hyperplane, we are looking for an integer $d$ such that $dH-\Lambda$ is effective. The question corresponds to ask if $d$ is bounded if the degree of $\Lambda$ is.

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$ such that the restriction of this linear system to $X$ yields $\Lambda$, plus some fixed components.

Is it true that the degree of $\Lambda'$ is bounded if the degree of $\Lambda$ is bounded? (the degree is here the degree in $\mathbb{P}^n$).

Comment: Denoting by $D\in \mathrm{Pic}(X)$ the divisor of $\Lambda$ and by $H\in \mathrm{Pic}(X)$ the very ample divisor given by the restriction of an hyperplane, we are looking for an integer $d$ such that $dH-\Lambda$ is effective. The question corresponds to ask if $d$ is bounded if the degree of $\Lambda$ is.

EDIT: The bound asked here can depend on $X$.

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$ such that the restriction of this linear system to $X$ yields $\Lambda$, plus some fixed components.

Is it true that the degree of $\Lambda'$ is bounded if the degree of $\Lambda$ is bounded? (the degree is here the degree in $\mathbb{P}^n$).

Comment: Denoting by $D\in \mathrm{Pic}(X)$ the divisor of $\Lambda$ and by $H\in \mathrm{Pic}(X)$ the very ample divisor given by the restriction of an hyperplane, we are looking for an integer $d$ such that $dH-\Lambda$ is effective. The question corresponds to ask if $d$ is bounded if the degree of $\Lambda$ is.

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$ such that the restriction of this linear system to $X$ yields $\Lambda$, plus some fixed components.

Is it true that the degree of $\Lambda'$ is bounded if the degree of $\Lambda$ is bounded? (the degree is here the degree in $\mathbb{P}^n$).

Comment: Denoting by $D\in \mathrm{Pic}(X)$ the divisor of $\Lambda$ and by $H\in \mathrm{Pic}(X)$ the very ample divisor given by the restriction of an hyperplane, we are looking for an integer $d$ such that $dH-\Lambda$ is effective. The question corresponds to ask if $d$ is bounded if the degree of $\Lambda$ is.