Ampleness of a divisor on a curve

On a smooth projective curve, positive degree is ample, degree $2g$ kills base points, and degree $2g+1$ is very ample.Suggest a slogan

Let $D$ be a divisor on a smooth projective curve $X$ of genus $g$ over $k = \bar{k}$.

$D$ is ample if and only if $\deg D > 0$.
$\deg D \geq 2g$ implies $\abs{D}$ is base-point free.
$\deg D \geq 2g+1$ implies $D$ is very ample.

On a curve ampleness collapses to a single integer, and this is the cleanest place to see the ample-against-very-ample distinction: positivity of the degree is ample, but very ampleness needs the degree to clear a genus-dependent threshold. Raising the degree past $2g+1$ is exactly what taking powers achieves, which is the definition of ample in this case.

The thresholds are what Riemann–Roch gives: once $\deg D > 2g-2$ the correction term vanishes and $h^0$ is computable, so the conditions for base points and for separation become arithmetic. Both bounds are sharp on an elliptic curve, where $g = 1$ and degree $2$ gives the base-point-free double cover of $\PP^1$ while degree $3$ first embeds.

When is a divisor on a curve ample?

What degree makes a divisor on a curve base-point free? Very ample?

Dependencies

D-DIVAMPLEAmple and very ample divisorsuses
PR-Y5S7VThe degree of a divisor on a projective curveuses

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Positivity