Let ${ℳ}_g$ be the moduli space of smooth complex projective curves of genus $g$. Here we prove that the subset of ${ℳ}_g$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${ℳ}_g$. As an application we show that if $X\in {ℳ}_g$ is defined over $ℝ$, then there exists a low degree pencil $uX\to {\mathbb{P}}^1$ defined over $ℝ$.

We study the lowest dimensional open case of the question whether every arithmetically Cohen–Macaulay subscheme of ${\mathbb{P}}^N$ is glicci, that is, whether every zero-scheme in ${\mathbb{P}}^3$ is glicci. We show that a general set of $n\ge 56$ points in ${\mathbb{P}}^3$ admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in ${\mathbb{P}}^3$.

We provide a simple characterization of codimension two submanifolds of ${\mathbb{P}}^n\left(ℝ\right)$ that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when $n\ge 6$. If the codimension two submanifold is a nonsingular algebraic subset of ${\mathbb{P}}^n\left(ℝ\right)$ whose Zariski closure in ${\mathbb{P}}^n\left(ℂ\right)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in ${\mathbb{P}}^n\left(ℝ\right)$.

We give examples of failure of the existence of co-fibered products in the category of algebraic curves.

Let $(R,𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\mathrm{mAss}}_R(R/I)={\mathrm{Assh}}_R\left(I\right)$. It is shown that the $R$-module $H_I^{\mathrm{ht}\left(I\right)}\left(R\right)$ is $I$-cofinite if and only if $\mathrm{cd}(I,R)=\mathrm{ht}\left(I\right)$. Also we present a sufficient condition under which this condition the $R$-module $H_I^i\left(R\right)$ is finitely generated if and only if it vanishes.

Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules $To{r}_i^R(N,M)$ are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules $To{r}_i^R(N,H_I^j\left(M\right))$ are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules $To{r}_i^R(N,M)$ are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it is shown that...