For smooth minimal surfaces of general type with pg >=4 and odd-degree canonical map onto a ruled surface, pg <= d+2, the image is a cone over a rational normal curve, and equality holds only for d=3,9,11; for d=5, pg<=5 with a specific pencil.
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Surfaces with canonical map of odd degree
For smooth minimal surfaces of general type with pg >=4 and odd-degree canonical map onto a ruled surface, pg <= d+2, the image is a cone over a rational normal curve, and equality holds only for d=3,9,11; for d=5, pg<=5 with a specific pencil.