On trigonal non-Gorenstein curves with zero Maroni invariant. (English) Zbl 1060.14036

Let \(C\) be a curve i.e. an integral one-dimensional scheme of finite type and complete over an algebraically closed field \(k\). \(C\) is called trigonal if it admits a pencil \(g_{3}^{1}\) and \(3\) is the smallest such degree. If \(C\) is nonhyperelliptic of genus \(g\) and \(\omega\) is a canonical divisor of \(C\) then the curve \(C^{\prime}= \varphi_{\omega} (\widetilde{C})\subset\mathbb{P}^{g-1}\) is called the canonical model of \(C\) (\(\widetilde{C}\) is the nonsingular model of \(C\) and \(\varphi_{\omega}\) is the canonical morphism associated to the linear series \(\left| \omega\right| \)). Using canonical models the author answer some questions concerning trigonal non-Gorenstein curves with zero Maroni invariant. The latter invariant is defined as the smallest Maroni invariants among all pencils \(g_{3}^{1}\) on \(C.\) This, in turn, is defined as \(m\) such that there exists \(n\geq m,\) \(m+n=g-2,\) for which \[ H^{0}(C,\omega)=\langle1,x,\ldots,x^{n},y,xy,\ldots,x^{m}y\rangle \] for \(x\in H^{0}(C,\mathfrak{a})\setminus k\) and \(y\in H^{0}(C,\omega).\) Among results we have for \(g\geq4\):
1. every trigonal curve with zero Maroni invariant is almost Gorenstein with at most one non-Gorenstein point,
2. if \(C\) is a Kunz curve of genus \(g\geq5\) then \(C\) is trigonal curves with zero Maroni invariant if and only if the canonical model \(C^{\prime}\) lies on a cone \(S\subset\mathbb{P}^{g-1},\)
3. a curve \(C,\) unibranch in all of its points, is trigonal with basic hyperelliptic canonical model if and only if there exists a point \(P\in C\) with a semigroup of values \(S_{P}=\{0,g,g+1,\ldots,2g-\eta-2,2g-\eta ,\rightarrow\}\) for some integer \(\eta\) such that \(1\leq\eta\leq g-3.\)


14H20 Singularities of curves, local rings
14H51 Special divisors on curves (gonality, Brill-Noether theory)
Full Text: DOI


[1] Ballico, E., Maximal degree subsheaves of torsion free sheaves on singular projective curves, Trans. amer. math. soc., 353, 9, 3617-3627, (2001) · Zbl 0977.14017
[2] Ballico, E., On the reflexivity of canonical models of non-Gorenstein curves, Internat. math. J., 1, 4, 363-365, (2002) · Zbl 1005.14013
[3] Ballico, E., Trigonal Gorenstein curves and Weierstrass points, Tsukuba J. math., 26, 133-144, (2002) · Zbl 1059.14502
[4] Barucci, V.; D’Anna, M.; Fröberg, R., Analytically unramified one-dimensional semilocal rings and their value semigroups, J. pure appl. algebra, 147, 215-254, (2000) · Zbl 0963.13021
[5] Barucci, V.; D’Anna, M.; Fröberg, R., The semigroup of values of a one-dimensional local ring with two minimal primes, Comm. algebra, 28, 8, 3607-3633, (2000) · Zbl 0964.13013
[6] Barucci, V.; Fröberg, R., One-dimensional almost Gorenstein rings, J. algebra, 188, 418-442, (1997) · Zbl 0874.13018
[7] Grothendieck, A.; Dieudonné, J., Élements de Géométrie algebrique I, () · Zbl 0203.23301
[8] Eisenbud, D.; Koh, J.; Stillmann, M., Determinantal equations for curves of high degree, Amer. J. math., 110, 513-539, (1988), (appendix with J. Harris) · Zbl 0681.14027
[9] Maroni, A., Le serie lineare speciale sulle curve trigonali, Ann. mat. pura appl., 25, 333-341, (1946)
[10] Rosenlicht, M., Equivalence relations on algebraic curves, Ann. of math., 56, 169-191, (1952) · Zbl 0047.14503
[11] Rosa, R., Non-classical trigonal curves, J. algebra, 225, 359-380, (2000) · Zbl 1020.14009
[12] Rosa, R.; Stöhr, K.-O., Trigonal Gorenstein curves, J. pure appl. algebra, 174, 187-205, (2002) · Zbl 1059.14038
[13] Serre, J.-P., Groupes algebriques et corps de classes, (1959), Hermann Paris · Zbl 0097.35604
[14] Stöhr, K.-O., On the poles of regular differentials of singular curves, Bol. soc. brasil. mat., 24, 105-135, (1993) · Zbl 0788.14020
[15] Stöhr, K.-O., Hyperelliptic Gorenstein curves, J. pure appl. algebra, 135, 93-105, (1999) · Zbl 0940.14018
[16] Stöhr, K.-O., Local and global zeta-functions of singular algebraic curves, J. number theory, 71, 172-202, (1998) · Zbl 0940.14016
[17] Stöhr, K.-O.; Viana, P., Weierstrass gap sequences and moduli varieties of trigonal curves, J. pure appl. algebra, 81, 63-82, (1992) · Zbl 0768.14016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.