| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

2. Measuring positivity of cycles

It is interesting to try to measure the positivity of a numerical cycle class.
    1. Mobility and mobility counts

          Let $X$ be an $n$ dimensional variety. Given an effective integral $k$-cycle $\alpha\in N_k(X)_\mathbb{Z}$, the mobility count of $\alpha$, denoted $mc(\alpha)$, is the maximum number of general points in $X$ that we can impose on an effective cycle of class $\alpha$. For example, any two points in $\mathbb{P}^2$ can be connected by a line $\ell \subset \mathbb{P}^2,$ so we see $mc([\ell])\ge 2$.

      The mobility count is supposed to be an analogue of $\text{dim}(H^0(X,\mathcal{O}(E)))$. Taking a cue from divisor theory it is natural to consider asymptotic invariants of a numerical cycle. Define the mobility of a numerical cycle $\alpha \in N_k(X)_\mathbb{Z}$ class to be $$\text{mob}(\alpha):= \frac{n! mc(m\cdot\alpha)}{m^{n/(n-k)}}.$$ Define the Iitaka dimension of $\alpha$ of a numerical cycle to be $$K(\alpha)=\text{max}\{ r \in\mathbb{R} | \text{limsup}_{m\rightarrow \infty} \frac{mc(m\cdot \alpha)}{m^r}>0 \}.$$

      Problem 2.1.

      Let $[\ell]\in N_1(\mathbb{P}^3)_\mathbb{Z}$ be the class of a line. What is $\text{mob}([\alpha])$?
        1. Remark. Complete intersections should give the optimal mobility.
            • Problem 2.2.

              What is $K(\alpha)$ for $\alpha \in N_k(\text{Gr}(k,n))$ a Schubert cycle?
                • Problem 2.3.

                  Develop better estimates for the growth rate of $mc(\alpha)$ for $\alpha$ a numerical class on $\text{Gr}(2,4)$ or $\mathbb{P}^2\times \mathbb{P}^2$.
                    • Problem 2.4.

                      Is $K(\alpha)$ an integer?
                        1. Remark. This is true for divisors, curves, $\text{Gr}(2,n)$, but is open in general.
                            • Other measures of positivity

                                  Let $X$ be a projective variety. Suppose that there is a class $\alpha \in \text{Eff}_k(X)_{\mathbb{Z}}$ such that for a general point $x\in X$ and a general $k$-plane $V\subset T_x X$ there is an irreducible subvariety $Y\subset X$ with $[V]= \alpha$ and $T_xY = V$.

                              Problem 2.5.

                              [Voisin] Is $\alpha$ in the interior of the effective cone?

                                  Cite this as: AimPL: Positivity of cycles, available at http://aimpl.org/poscycles.