We determine the gonality and the Clifford index for curves on a compact smooth toric surface. Moreover, it is shown that their gonality are computed by pencils on the ambient surface. From the geometrical view point, this means that the gonality can be read off from the lattice polygon associated to the curve.

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The spintronic properties of magnetic molecules have attracted significant scientific attention. Special emphasis has been placed on the qubit for quantum information processing. The single-molecule magnet bis(phthalocyaninato (Pc)) Tb(III) (TbPc2) is one of the best examined cases in which the delocalized π-radical electron spin of the Pc ligand plays the key role in reading and intermediating the localized Tb spin qubits. We utilized the electron spin resonance (ESR) technique implemented on a scanning tunneling microscope (STM) and use it to measure local the ESR of a single TbPc2 molecule decoupled from the Cu(100) substrate by a two-monolayer NaCl film to identify the π-radical spin. We detected the ESR signal at the ligand positions under the resonance condition expected for an S = 1/2 spin. The results reveal that the π-radical electron is delocalized within the ligands and exhibits intramolecular coupling susceptible to the chemical environment.

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