On the \(K_1\)-groups of algebraic curves (Q1401440)

The purpose of this paper is to study \(K_1\)-groups of algebraic curves, in particular the Block conjecture on it. For \(C\) a nonsingular projective curve over the complex number field \(\mathbb{C}\), the author studies \(K_1\) of the curve by using arithmetic Hodge structure. First, the author reviews admissible variation of mixed Hodge structure and calculates the Yoneda extension groups. Then he constructs the regulator map \[ \rho^2_C: V(C)\to \text{Ext}^2_{{\mathcal M}(C)}(Z(0),\;H^1(C,Z(2)) \] [a supplement of the construction of \textit{M. Asakura}, CRM Proc. Lect. Notes 24, 133-154 (2000; Zbl 0966.14014)]. It is also given a survey of the theory of generalized Jacobian rings with the theorem on Mori's connectivity for open curves. This result and the Mordell-Weil theorem over function fields are used to prove the main theorems in this paper: Theorem 1.2: Let \(C\subset\mathbb{P}^2_{\mathbb{C}}\) be a generic smooth plane curve of degree \(d\geq 4\), and \(\{P_1,\dots, P_n\}\) a set of closed points of \(C\) in a generic position. Then the map \[ \bigoplus^{n-1}_{i=1} \mathbb{C}^*\otimes [P_i- P_{i+1}]\to \text{Ext}^2_{{\mathcal M}(C)}(Z(0), H^1(C,Z(2))) \] induced from \(\rho^2_C\) is injective modulo torsion. Theorem 1.4: Let \(C\subset\mathbb{P}^2_\mathbb{C}\) be a generic plane curve of degree \(d\geq 4\). Then the map \[ \overline \mathbb{Q}^*\otimes \text{Pic}^0(\mathbb{C})\to \text{Ext}^2_{{\mathcal M}(C)}(Z(0), H^1(C,Z(2))) \] induced from \(\rho^2_C\) is injective.