# Application of the geometric version of the Riemann-Roch.

In the book "Phillips Griffiths and Joe Harris, Principles of Algebraic Geometry", page 341, last paragraph, the following result is stated:

"Our first object is to describe geometrically the tangent cone to the variety $W_d$ at a point $\mu(D)$. Suppose first that the divisor $D$ is regular, i.e., that dim$|D|=0$. By the geometric version of the Riemann-Roch, the linear span $\overline{D}$ of the points of $D$ on the canonical curve is a $(d-1)$-plane, and we claim that:

$W_d$ is smooth at $\mu(D)$, with tangent space $T_{\mu(D)}(W_d)= \overline{D}$."

Proof of this assertion is given for two cases. Suppose first that $D=p_1+...+p_d$ with the points $p_i$ distinct (...) and then the statement is completed.

In a second moment of the proof, it is assumed that $D=2p_1+...+p_d$ where $p_i$ are distinct. After some manipulation in the neighborhood of point $p_1$, the argument proceeds as before when the points were distinct.

My question is therefore directed to the following part of the statement: "(...) the linear span $\overline{D}$ of the points of $D$ on the canonical curve is a $(d-1)$-plane"

Once $D=p_1+...+p_d$ with the points $p_i$ distinct, it is clear that linear span $\overline{D}$ is a $(d-1)$-plane. But, being $D=2p_1+...+p_d$ where $p_i$ are distinct, then dim$\overline{D}< (d-1)$.

And then, I get confused by the statement. It is then correct to think (through the text) that:

$W_d$ is smooth at $\mu(D)$, with tangent space $T_{\mu(D)}(W_d)= \overline{D}$, and dim $T_{\mu(D)}(W_d)=d-1$, for $D=p_1+...+p_d$ with the points $p_i$ distinct and dim $T_{\mu(D)}(W_d)=d-2$, for $D=2p_1+...+p_d$ with the points $p_i$ ???

Thank you.