  We explore the geometry behind the modular bootstrap and its image in the
space of Taylor coefficients of the torus partition function. In the first
part, we identify the geometry as an intersection of planes with the convex
hull of moment curves on $R^+{\otimes}\mathbb{Z}$, with boundaries
characterized by the total positivity of generalized Hankel matrices. We phrase
the Hankel constraints as a semi-definite program, which has several
advantages, such as constant computation time with increasing central charge.
We derive bounds on the gap, twist-gap, and the space of Taylor coefficients
themselves. We find that if the gap is above $\Delta^*_{gap}$, where
$\frac{c{-}1}{12}<\Delta^*_{gap}< \frac{c}{12}$, all coefficients become
bounded on both sides and kinks develop in the space. In the second part, we
propose an analytic method of imposing the integrality condition for the
degeneracy number in the spinless bootstrap, which leads to a non-convex
geometry. We find that even at very low derivative order this condition rules
out regions otherwise allowed by bootstraps at high derivative order.
