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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 $$1\leq\eta\leq g-3.$$

##### MSC:
 14H20 Singularities of curves, local rings 14H51 Special divisors on curves (gonality, Brill-Noether theory)
##### Keywords:
trigonal curve; Gorenstein curve; canonical model; base point
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