A curve \(X\) of genus \(g\) has a nonspecial divisor \(D\) of degree \(d\) such that \(\abs{D}\) has no base points if and only if \(d \geq g+1\).
(Erratum)
As printed, there is one exceptional case. If \(g=0\), then \(X\cong\PP^1\) and the zero divisor is nonspecial and has base-point-free complete linear system, but its degree is \(0<g+1\).
Thus the stated equivalence is correct for \(d>0\). Without the positivity assumption, the complete statement is that such a divisor exists exactly when \[ d\ge g+1 \] or \[ (g,d)=(0,0). \]
Solution
Write \[ L=\OO_X(D). \]
1. If \(D\) is nonspecial and \(\abs{D}\) is base-point free, then either\[ (g,d)=(0,0) \]or\[ d\ge g+1. \]
Nonspeciality means \[ h^1(X,L)=0. \] Hence Riemann–Roch gives \[ h^0(X,L)=d+1-g. \]
Suppose first that \[ h^0(X,L)=1. \] Because \(\abs{D}\) has no base points, its unique nonzero section vanishes nowhere. It therefore trivializes \(L\), so \[ L\cong\OO_X, \qquad d=0. \] Nonspeciality now says \[ 0=h^1(X,\OO_X)=g, \] so \((g,d)=(0,0)\).
Outside this exceptional case one has \[ h^0(X,L)\ge2. \] Using Riemann–Roch again, \[ d+1-g=h^0(X,L)\ge2, \] and therefore \[ d\ge g+1. \]
2. If \(g=0\) and \(d\ge1\), then there is a nonspecial base-point-freedivisor of degree \(d\).
For a smooth projective genus-zero curve over the algebraically closed ground field, \[ X\cong\PP^1. \] Take \[ L=\OO_{\PP^1}(d). \] For \(d\ge0\) this line bundle is generated by its global sections, so its complete linear system has no base points. Also \[ h^1(\PP^1,\OO(d))=0 \] for \(d\ge-1\), hence in particular for \(d\ge1\). Any divisor \(D\) with \(\OO_X(D)\cong L\) has the required properties.
3. Assume \(g\ge1\) and fix an integer\[ d\ge g+1. \]Then \(\Pic^d(X)\) is an irreducible variety of dimension \(g\).
Fix a point \(P_0\in X\). Tensoring by \[ \OO_X(dP_0) \] identifies \[ \Pic^0(X) \xrightarrow{\sim} \Pic^d(X). \] By T-CRVJACFUN, \[ \Pic^0(X)=\Jac(X) \] is an irreducible variety of dimension \(g\). Hence the same is true of \(\Pic^d(X)\).
4. The locus\[ S_d = \{L\in\Pic^d(X):h^1(X,L)>0\} \]of special line bundles is contained in a proper closed subset of\(\Pic^d(X)\).
By Serre duality, \[ h^1(X,L)=h^0(X,K\otimes L^{-1}). \] Thus \(L\) is special exactly when \[ K\otimes L^{-1} \] is effective.
Put \[ e=2g-2-d. \] If \(e<0\), there are no such line bundles and \(S_d\) is empty.
If \(e\ge0\), every special \(L\) has the form \[ L\cong\OO_X(K-E) \] for some effective divisor \(E\) of degree \(e\). Hence \(S_d\) is the image of the proper morphism \[ \Sym^eX \longrightarrow \Pic^d(X), \qquad E\longmapsto\OO_X(K-E). \] It is therefore closed, and \[ \dim S_d\le e=2g-2-d. \] Since \(d\ge g+1\), \[ 2g-2-d\le g-3<g. \] Thus \(S_d\) is proper in the \(g\)-dimensional variety \(\Pic^d(X)\).
5. The locus of line bundles in \(\Pic^d(X)\) having a base point iscontained in a proper closed subset.
Let \(L\in\Pic^d(X)\) and suppose that a point \(P\in X\) is a base point of \(\abs{L}\). Then \[ h^0(X,L(-P))=h^0(X,L). \] Riemann–Roch for \(L\) and \(L(-P)\) gives \[ h^0(X,L) = d+1-g+h^0(X,K\otimes L^{-1}) \] and \[ h^0(X,L(-P)) = d-g+h^0(X,K\otimes L^{-1}(P)). \] Equality therefore implies \[ h^0(X,K\otimes L^{-1}(P)) = h^0(X,K\otimes L^{-1})+1 \ge1. \] Hence \[ K\otimes L^{-1}(P) \] is effective.
Put \[ e'=2g-1-d. \] If \(e'<0\), this is impossible, so every degree-\(d\) line bundle is base-point free.
If \(e'\ge0\), every line bundle having a base point is contained in the image of \[ X\times\Sym^{e'}X \longrightarrow \Pic^d(X), \qquad (P,E)\longmapsto\OO_X(K+P-E). \] The source is proper, so its image \(B_d\) is closed. Moreover, \[ \dim B_d \le 1+e' = 2g-d \le g-1 <g. \] Thus \(B_d\) is a proper closed subset of \(\Pic^d(X)\) containing every line bundle with a base point.
6. If \(g\ge1\) and \(d\ge g+1\), there exists a nonspecial line bundle\(L\) of degree \(d\) whose complete linear system has no base points.
By step 3, \(\Pic^d(X)\) is irreducible of dimension \(g\). Steps 4 and 5 give two proper closed subsets \[ S_d, \qquad B_d. \] Their union is still a proper closed subset of the irreducible variety \(\Pic^d(X)\). Choose \[ L\in\Pic^d(X)\setminus(S_d\cup B_d). \] Then \(L\notin S_d\) makes \(L\) nonspecial, and \(L\notin B_d\) implies that \(\abs{L}\) has no base points.
Choose a divisor \(D\) with \[ \OO_X(D)\cong L. \] Then \[ \deg D=d, \] and \(D\) has the required properties.
7. For positive degree \(d\), a nonspecial divisor of degree \(d\) withbase-point-free complete linear system exists if and only if\[ \boxed{d\ge g+1}. \]
For the forward implication, positive degree rules out the exceptional case in step 1, so that step gives \(d\ge g+1\).
For the reverse implication, step 2 proves existence when \(g=0\), and step 6 proves it when \(g\ge1\).
8. Without the positivity assumption, the only additional case is\[ \boxed{(g,d)=(0,0)}. \]
The zero divisor on \(\PP^1\) supplies this case, as recorded in the erratum. Step 1 shows that no other case with \(d<g+1\) can occur.
9. Q.E.D. for the corrected statement.
Steps <1>7–<1>8 give exactly the corrected classification stated in the erratum.