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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)
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