A Riemann singularity theorem for integral curves. (English) Zbl 1254.14032
The authors study “the local geometry of the theta divisor of an integral curve with the goal of extending the Riemann Singularity Theorem”. The paper is partially motivated by a recent paper of L. Caporaso [J. Eur. Math. Soc. (JEMS) 11, No. 6, 1385–1427 (2009; Zbl 1202.14030)]. The main result of the paper is the following:
Theorem A. Suppose that \(X/k\) is an integral curve with at worst planar singularities. Let \(x\) be a point of the theta divisor \(\Theta\) corresponding to rank 1, torsion-free sheaf \(I\). If the sheaf \(I\) fails to be locally free at \(n\) nodes and no other points, then the multiplicity and order of vanishing of \(\Theta\) at \(x\) satisfy the equation \[ \text{mult}_x\Theta =(\text{mult}_x \overline{J} ^{g-1}_{X/k})\cdot \text{ord}_x \Theta = 2^n\cdot h^0(X,I). \] In this theorem \(\overline{J}^{g-1}_{X/k}\) is the completion of the moduli space of line bundles of degree \(g-1\) on the curve \(X/k\) and it is the moduli space of torsion-free sheaves of rank 1. When \(I\) is a line bundle the result was obtained in [G. R. Kempf, The singularities of certain varieties in the Jacobian of a curve, Ph.D. thesis, Columbia University, New York (1970)].
The paper suggests also a “formula for the multiplicity of the theta divisor of a singular, integral curve at a point and present some evidence that this formula should hold”.
The relations with the existing important literature are presented and some suggestions for further research are discussed.
Theorem A. Suppose that \(X/k\) is an integral curve with at worst planar singularities. Let \(x\) be a point of the theta divisor \(\Theta\) corresponding to rank 1, torsion-free sheaf \(I\). If the sheaf \(I\) fails to be locally free at \(n\) nodes and no other points, then the multiplicity and order of vanishing of \(\Theta\) at \(x\) satisfy the equation \[ \text{mult}_x\Theta =(\text{mult}_x \overline{J} ^{g-1}_{X/k})\cdot \text{ord}_x \Theta = 2^n\cdot h^0(X,I). \] In this theorem \(\overline{J}^{g-1}_{X/k}\) is the completion of the moduli space of line bundles of degree \(g-1\) on the curve \(X/k\) and it is the moduli space of torsion-free sheaves of rank 1. When \(I\) is a line bundle the result was obtained in [G. R. Kempf, The singularities of certain varieties in the Jacobian of a curve, Ph.D. thesis, Columbia University, New York (1970)].
The paper suggests also a “formula for the multiplicity of the theta divisor of a singular, integral curve at a point and present some evidence that this formula should hold”.
The relations with the existing important literature are presented and some suggestions for further research are discussed.
Reviewer: Nicolae Manolache (Bucureşti)
