# Calculate Kodaira dimension of a singular hypersurface

For a smooth projective hypersurface $$H \subseteq \mathbb{P}^n$$ of degree $$d$$ one can calculate its Kodaira dimension $$\kappa(H)$$, and find $$\kappa(H) = \begin{cases} -\infty \qquad &\mbox{if } d < n +1,\\ 0 &\mbox{if } d = n+1,\\ \dim H &\mbox{if } d > n+1. \end{cases}$$

What about if $$H$$ is not smooth? Can we say anything about its Kodaira dimension, or even sensibly calculate something we can call the ''Kodaira dimension''?

I've only ever seen canonical divisors defined over smooth varieties, so am unsure how to proceed in the singular case.

• One just defines the Kodaira dimension as the dimension of a resolution of singularities, because it is known to be birationally invariant, so it does not depend on a choice of resolution. The answer obviously depends a lot on the type of singularities, since every variety is birational to a hypersurface. But I'm sure there are nice formulas in many interesting special cases... – Will Sawin Mar 2 at 16:26
• As long as your varieties have canonical singularities you can compute the Kodaira dimension the same way as in the smooth case. For the definition of the canonical sheaf/divisor see mathoverflow.net/questions/35736/…. – Sándor Kovács Mar 4 at 15:54
• There is also an interesting computation of the Kodaira dimension of a singular variety on Terry Tao's blog: terrytao.wordpress.com/2014/12/20/…, see also the last comment for a more AG way of the same: terrytao.wordpress.com/2014/12/20/… – Sándor Kovács Mar 4 at 15:54