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Brill-Noether theory for curves of a fixed gonality. (English) Zbl 1456.14038

Authors’ abstract: “We prove a generalisation of the Brill-Noether theorem for the variety of special divisors \(W^r_d(C)\) on a general curve \(C\) of prescribed gonality. Our main theorem gives a closed formula for the dimension of \(W^r_d(C)\). We build on previous work of N. Pflueger [Adv. Math. 312, 46–63 (2017; Zbl 1366.14031)], who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne [D. Cartwright et al., Can. Math. Bull. 58, No. 2, 250–262 (2015; Zbl 1327.14266)]. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of D. E. Speyer [Algebra Number Theory 8, No. 4, 963–998 (2014; Zbl 1301.14035)] on genus \(1\) curves to arbitrary genus.”
More precisely, the main result is the following:
Theorem. Let \(C\) be a general curve of genus \(g\) and gonality \(k\ge 2\) over the complex numbers. Assume that the quantity \(g-d+r\) is positive. Then \[\dim W^r_d(C)=\max_{\ell\in\{0,\dots,r'\}} g-(r-\ell+1)(g-d+r-\ell)-\ell k\] where \(r'=\min\{r, g-d + r-1\}\).

MSC:

14H51 Special divisors on curves (gonality, Brill-Noether theory)
14T15 Combinatorial aspects of tropical varieties

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References:

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