OFFSET
0,3
COMMENTS
a(g) counts the combinatorial types of hyperelliptic stable curves of genus g over an algebraically closed field of characteristic != 2; equivalently, over C with the complex topology, this is the number of homeomorphism classes.
a(g) is also the number of stable hyperelliptic graphs of genus g (for the definition, see Kawaguchi and Yamaki).
a(g) is also the number of cluster pictures for 2g+2 points (for the definition, see Dokchitser et al.).
LINKS
Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, and Adam Morgan, Semistable types of hyperelliptic curves, arXiv:1704.08338 [math.NT], 2017; Algebraic curves and their applications, Contemp. Math. 724 (2019), 73-135. DOI: 10.1090/conm/724/14586.
Shu Kawaguchi and Kazuhiko Yamaki, Rank of divisors on hyperelliptic curves and graphs under specialization, arXiv:1304.6979 [math.AG], 2013-2014; Int. Math. Res. Notices, 12 (2015), 4121-4176. DOI: 10.1093/imrn/rnu059.
Max Schwegele, Hyperelliptic Stable Curves, arXiv:2606.29016 [math.AG], 2026.
EXAMPLE
a(0) = a(1) = 0 because there are no stable curves of genus 0 or 1.
a(2) = A174224(2) = 7 because every stable curve of genus 2 is automatically hyperelliptic.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Max Schwegele, Jul 09 2026
STATUS
approved