Jaewoo Jeong, Hankel Index of Smooth Non-ACM Curves of Almost Minimal Degree

 

The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. Note that the Hankel index of a variety is difficult to compute and was computed for just few cases. In 2017, Blekherman, Sinn, and Velasco provided an captivating (lower) bound of the Hankel index of a variety by an algebraic invariant, Green-Lazarsfeld index, of the variety. In particular, if the variety X is an arithmetrically Cohen-Macaulay (ACM) variety of almost minimal degree, then the Hankel index of X equals to the Green-Lazarsfeld index of X plus one (which is the equality case of the bound). We study the Hankel index of smooth non-ACM curves of almost minimal degree. Note that the curve is the image of the projection of rational normal curves away from an outer point. It is known that the Green-Lazarsfeld index of the curve is determined by the rank of the center of the projection with respect to the rational normal curve. We found a new rank of the center that detects the Hankel index of the rational curves. In addition, it turns out that the rational curves are the first class of examples that the lower bound of the Hankel index is strict.