Definition 19.6.1ACD722F45231

Definition 19.6.1.ACD722F45231 An Okounkov test curve relative to $\Delta $ consists of

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   a number $\Delta _{\max }\in \mathbb {R}$ and

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   an assignment $(-\infty ,\Delta _{\max })\ni \tau \mapsto \Delta _{\tau }\in \mathcal{K}_ n$ satisfying

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      the assignment $\tau \mapsto \Delta _{\tau }$ is decreasing and concave¹;

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      we have $\Delta _{\tau }\xrightarrow {d_{\mathrm{Haus}}}\Delta $ as $\tau \to -\infty $.

The set of Okounkov test curves relative to $\Delta $ is denoted by $\mathrm{TC}(\Delta )$.

An Okounkov test curve $\Delta _{\bullet }$ relative to $\Delta $ is bounded if $\Delta _{\tau }=\Delta $ when $\tau $ is small enough. The subset of bounded Okounkov test curves is denoted by $\mathrm{TC}^{\infty }(\Delta )$.

An Okounkov test curve $\Delta _{\bullet }$ relative to $\Delta $ is said to have finite energy if

The subset of Okounkov test curves with finite energy is denoted by $\mathrm{TC}^{1}(\Delta )$.

Given $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$ and $\Delta '_{\bullet }\in \mathrm{TC}(\Delta ')$, we say $\Delta _{\bullet }\leq \Delta '_{\bullet }$ if $\Delta _{\max }\leq \Delta '_{\max }$ and for any $\tau <\Delta _{\max }$, we have $\Delta _{\tau }\subseteq \Delta '_{\tau }$.