Positivity of cycles

    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 $$K(\alpha)=\text{max}\{ r \in\mathbb{R} | \text{limsup}_{m\rightarrow \infty} \frac{mc(m\cdot \alpha)}{m^r}>0 \}.$$

    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 $T_xY = V$.