Non-trivial Linear Systems on Smooth Plane Curves

Non-trivial Linear Systems on Smooth Plane Curves

Let C be a smooth plane curve of degree d defined over an algebraically closed field k. In [10], while studying space curves, Max Noether considered the following question. For n ∈ Z≥1 find l(n) ∈ Z≥0 such that there exists a linear system g l(n) n on C but no linear system g l(n)+1 n and classify those linear systems g l(n) n on C. The arguments given by Noether in the answer to this question ...

Linear Systems of Plane Curves

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Non-primitive Linear Systems on Smooth Algebraic Curves and a Generalization of Maroni Theory

Let X be a smooth curve of genus g and M,L special spanned line bundles with h(X,L) = 2 and h(X,M ⊗ L∗) = h(X,M) − 2 > 0. Generalizing Maroni theory for trigonal curves we study the existence of such triples (X,M,L) for certain numerical invariants and classify all spanned line bundles, R, on any such X with h(X,M ⊗ R∗) > 0. We construct smooth curves with non-primitive special linear systems w...

Linear Systems on Tropical Curves

A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modul...