The degree thresholds $2g$ and $2g+1$

On a genus-$g$ curve, positive degree is ample, degree $2g$ frees base points, and degree $2g+1$ is very ample.Suggest a slogan

Let $D$ be a divisor on a smooth projective curve $C$ of genus $g$.

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

The last two are sufficient, not necessary.

All three come from running Riemann–Roch twice and watching the speciality term vanish. For base-point freeness one needs $\ell(D-p) = \ell(D)-1$; with $\deg D \geq 2g$ both $D$ and $D-p$ have degree $> 2g-2$, both are nonspecial, and subtracting the two formulas gives the drop of exactly one. For very ampleness one needs the drop by two at every pair, and $\deg D \geq 2g+1$ keeps $D-p-q$ nonspecial as well. Ampleness follows because $\deg nD > 0$ eventually exceeds $2g+1$.

The thresholds are not sharp, and the standard counterexample is the canonical divisor on a smooth plane quartic: $g=3$, $\deg K = 4 < 2g+1 = 7$, and $K$ is very ample. Standard specializations are: on $\PP^1$, ample, very ample, and $\deg D\ge1$ coincide; on an elliptic curve, $D$ is very ample exactly when $\deg D\ge3$, giving the plane cubic model; on a genus-$2$ curve, degree $5$ gives a quintic model in $\PP^3$. If $D$ is very ample and $\varphi$ is the resulting embedding, then $\deg \varphi(C) = \deg D$.

When is a divisor on a curve ample?

When is a divisor on a curve base-point free?

What degree guarantees a divisor on a curve is very ample?

Dependencies

T-MWDVLRiemann--Roch and Serre duality on a curveuses
D-CRVSPECSpecial divisors and the index of specialityuses

Backlinks

T-CRVEMBP3Every curve embeds in $\PP^3$ and projects to a nodal plane curveuses

What links to this

Curves in projective space