Regularity and Koszul property of symbolic powers of monomial ideals

Let I be a homogeneous ideal in polynomial ring over field. $$I^{(n)}$$ the n-th symbolic power of I. Motivated by results about ordinary powers I, we study asymptotic behavior regularity function $${{\,\mathrm{reg}\,}}(I^{(n)})$$ and maximal generating degree $$\omega (I^{(n)})$$ , when is monomial ideal. It known that both functions are eventually quasi-linear. We show that, addition, sequences $$\{{{\,\mathrm{reg}\,}}I^{(n)}/n\}_n$$ $$\{\omega (I^{(n)})/n\}_n$$ converge to same limit, which can described combinatorially. construct an example equidimensional, height two squarefree for not linear functions. For last goal, introduce new method establishing componentwise linearity ideals. This allows us identify class ideals whose linear.