# lineBundleFromPointsAndMultipliers -- Computes basis of a line bundle from the 2g points P_i, Q_i and the multipliers

## Synopsis

• Usage:
lineBundleFromPointsAndMultipliers(multL,P,Q,k)
• Inputs:
• multL, a list, containing the normalized multipliers a_i (normalized means that b_i=1)
• P, , the coordinate matrix of g points in P^1
• Q, , the coordinate matrix of g points in P^1
• k, an integer, defining the degree of the line bundle
• Outputs:
• f, , a basis of sections of the line bundle defined by the points P_i, Q_i and the multipliers

## Description

If C is a g-nodal canonical curve with normalization $\nu:\ P^1 \to P^{g-1}$ then a line bundle L of degree k on C is given by $\nu^*(O_{P^1}(k))\cong L$ and gluing data $\frac{b_j}{a_j}:O_{P^1}\otimes kk(P_j)\to O_{P^1}\otimes kk(Q_j)$. Given 2g points P_i, Q_i and the multipliers (a_i,b_i) we can compute a basis of sections of L as a kernel of the matrix $A=(A)_{ij}$ with $A_{ij}=b_iB_j(P_i)-a_iB_j(Q_i)$ where $B_j:P^1\to kk,\ (p_0:p_1)\to p_0^{k-j}p_1^j$.