We propose here a generalization of the problem addressed by the SHGH conjecture. The SHGH conjecture posits a solution to the question of how many conditions a general union $X$ of fat points imposes on the complete linear system of curves in $\mathbb{P}^{2}$ of fixed degree $d$, in terms of the occurrence of certain rational curves in the base locus of the linear subsystem defined by $X$. As a first step towards a new theory, we show that rational curves play a similar role in a special case of a generalized problem, which asks how many conditions are imposed by a general union of fat points on linear subsystems defined by imposed base points. Moreover, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, we relate our results to the failure of a strong Lefschetz property, and we give a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements.

A central problem in liaison theory is to decide whether every arithmetically Cohen–Macaulay subscheme of projective $n$-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can indeed be achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an $(n+ 1)$-dimensional projective space. For example, this result applies to all reduced arithmetically Cohen–Macaulay subschemes. We also show that every union of fat points in projective 3-space can be linked in the same space to a union of simple points in finitely many steps, and hence to a complete intersection in projective 4-space.

We study the Hilbert functions of fat points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. If $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function ${{H}_{Z}}(l,\,j)$ and ${{H}_{Z}}(i,\,l)$ eventually become constant for $l\,\gg \,0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. This enables us to compute all but a finite number values of ${{H}_{Z}}$ without using the coordinates of points. We also characterize the $\text{ACM}$ fat point schemes using our description of the eventual behaviour. In fact, in the case that $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is $\text{ACM}$, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Let $F$ be a divisor on the blow-up $X$ of ${{\mathbf{P}}^{2}}$ at $r$ general points ${{p}_{1}},...,{{p}_{r}}$ and let $L$ be the total transform of a line on ${{\mathbf{P}}^{2}}$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map ${{\mu }_{F}}:\Gamma ({{\mathcal{O}}_{_{X}}}(F))\otimes \Gamma ({{\mathcal{O}}_{_{X}}}(L))\to \Gamma ({{\mathcal{O}}_{_{X}}}(F)\otimes {{\mathcal{O}}_{_{X}}}(L))$ to the case that $F$ is ample. As an application, a formula for the dimension of the cokernel of ${{\mu }_{_{F}}}$ is obtained when $r\,=\,7$, completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes ${{m}_{1}}\,{{p}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{m}_{7}}\,{{p}_{7}}\,\subset \,{{\mathbf{P}}^{2}}$. All results hold for an arbitrary algebraically closed ground field $k$.