TY - JOUR
T1 - On the rank of general linear series on stable curves
AU - Christ, Karl
N1 - Publisher Copyright:
© The Author(s) 2023.
PY - 2024/2
Y1 - 2024/2
N2 - We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree \(d = g - 1\), we characterize when the effective locus gives a Theta divisor. In case of degree \(g - 2\) and \(g\), we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least \(2\). If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus contains an irrreducible component of expected dimension.
AB - We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree \(d = g - 1\), we characterize when the effective locus gives a Theta divisor. In case of degree \(g - 2\) and \(g\), we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least \(2\). If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus contains an irrreducible component of expected dimension.
KW - math.AG
KW - 14H20
KW - 14H40
KW - 14H51
UR - http://www.scopus.com/inward/record.url?scp=85147558672&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2005.12817
DO - 10.48550/arXiv.2005.12817
M3 - Article
VL - 388
SP - 2217
EP - 2240
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 2
ER -