A combinatorial interpretation for Schreyer's tetragonal invariants (Q273535)

In this paper, the authors consider the relation between combinatorial and geometric invariants for curves on toric surfaces. Previous works [\textit{W. Castryck} and \textit{F. Cools}, ``Linear pencils encoded in the Newton polygon'', Int. Math. Res. Notices 2017, No. 10, 2998--3049 (2017; \url{doi:10.1093/imrn/rnw082})], [\textit{A.G. Khovanskii}, Funct. Anal. Appl. 11, 289--296 (1978; Zbl 0445.14019)] showed that one can read some geometric invariants of these curves, such as the genus, the gonality or the Clifford index, out of the Newton polygon of the defining equation. Here, the authors show that this is true also for the Schreyer's invariants of a tetragonal curve [\textit{F.-O. Schreyer}, Math. Ann. 275, 105--137 (1986; Zbl 0578.14002)]: these are two integers \(b_1\) and \(b_2\) that determine the graded Betti numbers for the canonical model. Using this, they can to prove an intrinsicness result on Newton polygons of small lattice width.