Let$X$and$Y$be smooth varieties of dimensions$n$−1 and$n$over an arbitrary algebraically closed field,$f: X→Y$a finite map that is birational onto its image. Suppose that$f$is curvilinear; that is, for all$xεX$, the Jacobian ϱ$f(x)$has rank at least$n$−2. For$r$≥1, consider the subscheme$N$_{$r$}of$Y$defined by the ($r$−1)th Fitting ideal of the $$\mathcal{O}_Y $$ -module $$f_ * \mathcal{O}_X $$ , and set$M$_{$r$}∶=$f$^{−1}$N$_{$r$}. In this setting—in fact, in a more general setting—we prove the following statements, which show that$M$_{$r$}and$N$_{$r$}behave like reasonable schemes of source and target$r$-fold points of$f$.

Adult stem cells, which exist throughout the body, multiply by cell division to replenish dying cells or to promote regeneration to repair damaged tissues. To perform these functions during the lifetime of organs or tissues, stem cells need to maintain their populations in a faithful distribution of their epigenetic states, which are suscepti- ble to stochastic fluctuations during each cell division, unexpected injury, and potential genetic mutations that occur during many cell divisions. However, it remains unclear how the three processes of differentiation, proliferation, and apoptosis in regulating stem cells collectively manage these challenging tasks. Here, without consid- ering molecular details, we propose a genetic optimal control model for adult stem cell regeneration that includes the three fundamental processes, along with cell division and adaptation based on differ- ential fitnesses of phenotypes. In the model, stem cells with a distri- bution of epigenetic states are required to maximize expected performance after each cell division. We show that heteroge- neous proliferation that depends on the epigenetic states of stem cells can improve the maintenance of stem cell distributions to create balanced populations. A control strategy during each cell division leads to a feedback mechanism involving heterogeneous proliferation that can accelerate regeneration with less fluctuation in the stem cell population. When mutation is allowed, apoptosis evolves to maximize the performance during homeostasis after multiple cell divisions. The overall results highlight the importance of cross-talk between genetic and epigenetic regulation and the performance objectives during homeostasis in shaping a desirable heterogeneous distribution of stem cells in epigenetic states.

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This paper is devoted to exploiting the restrictions of Riesz–Morrey potentials on either unbounded or bounded domains in Euclidean spaces.