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Martin Sleziak
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Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first provedproved (Tommaso de Fernex, Mircea Mustata, Limits of log canonical thresholds) via non-standard methods:

  1. $T_n$ is closed in $\mathbb R$ for all $n$.

  2. The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emergedemerged later (J. Kollár, Which powers of holomorphic functions are integrable?), but the first one used non-standard technique.

Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first proved (Tommaso de Fernex, Mircea Mustata, Limits of log canonical thresholds) via non-standard methods:

  1. $T_n$ is closed in $\mathbb R$ for all $n$.

  2. The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emerged later (J. Kollár, Which powers of holomorphic functions are integrable?), but the first one used non-standard technique.

Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first proved (Tommaso de Fernex, Mircea Mustata, Limits of log canonical thresholds) via non-standard methods:

  1. $T_n$ is closed in $\mathbb R$ for all $n$.

  2. The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emerged later (J. Kollár, Which powers of holomorphic functions are integrable?), but the first one used non-standard technique.

fixed arxiv front-end link
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David Roberts
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Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first provedproved (Tommaso de Fernex, Mircea Mustata, Limits of log canonical thresholds) via non-standard methods:

  1. $T_n$ is closed in $\mathbb R$ for all $n$.

  2. The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emergedemerged later (J. Kollár, Which powers of holomorphic functions are integrable?), but the first one used non-standard technique.

Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first proved via non-standard methods:

  1. $T_n$ is closed in $\mathbb R$ for all $n$.

  2. The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emerged later, but the first one used non-standard technique.

Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first proved (Tommaso de Fernex, Mircea Mustata, Limits of log canonical thresholds) via non-standard methods:

  1. $T_n$ is closed in $\mathbb R$ for all $n$.

  2. The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emerged later (J. Kollár, Which powers of holomorphic functions are integrable?), but the first one used non-standard technique.

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Hailong Dao
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Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first proved via non-standard methods:

  1. $T_n$ is closed in $\mathbb R$ for all $n$.

  2. The set of points of accumulations from above of $T_n$ is $T_{n-1}$.

I think proofs that avoid non-standard analysis emerged later, but the first one used non-standard technique.