In this paper, we study the projective geometry of smooth models in $\Pro^4$ of Suzuki curves, employing the Weierstrass semigroup at the only singular point of the curves. In particular, we explicitly count the hypersurfaces of $\Pro^4$ containing the smooth projective model and provide a geometric characterization of those of small degree. We also prove that the characterization cannot be extended to higher-degree hypersurfaces of $\Pro^4$.

In this paper we study certain determinantal ideals that extend the class of ideals of Herzog-Northcott type introduced by O'Carroll and Planas-Vilanova. As is well known, in a three-dimensional Cohen-Macaulay local ring, the second symbolic powers of ideals of Herzog-Northcott type can be controlled well. We aim to generalize this fact considering ``saturation" instead of ``symbolic power." Furthermore, in order to compare the saturation with the symbolic power, we study the associated primes of the powers of certain determinantal ideals.

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-S\"oderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression of the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from \cite{ES1}. In higher codimension, obstructions arise that inspire our work on an alternative algorithm.

Let $s(n,q)$ be the smallest number $s$ such that any $n$-fold $\mathbb{ F}_q$-valued interpolation problem in $\mathbb{P}^k_{\mathbb{F}_q}$ has a solution of degree~$s$, that is: for any pairwise different $\mathbb{F}_q$-rational points $P_1,\ldots ,P_n$, there exists a hypersurface $H$ of degree~$s$ defined over $\mathbb{F}_q$ such that $P_1,\ldots ,P_{n-1}\in H$ and $P_n\notin H$. This function $s(n,q)$ was studied by Kunz and the second author in \cite{KuW} and completely determined for $q=2$ and $q=3$. For $q\geq 4$, we improve the results from \cite{KuW}.

The affine analogue to $s(n,q)$ is the smallest number $s=s_a(n,q)$ such that any $n$-fold $\mathbb{F}_q$-valued interpolation problem in $\mathbb{A}^k(\mathbb{F}_q)$, $k\in \mathbb{N}_{>0}$ has a polynomial solution of degree $\leq s$. We exactly determine this number.

We introduce a notion of degenerations of graded modules. In relation to it, we also introduce several partial orders as graded analogies of the hom order, the degeneration order and the extension order. We prove that these orders are identical on the graded Cohen-Macaulay modules over a graded ring which is of graded finite representation type and representation directed.

Let $T$ be a complete local ring with maximal ideal $M$, $C$ a countable set of incomparable prime ideals of $T$, and $B_1$ and $B_2$ sets of prime ideals of $T[[x_1,\ldots,x_n]]$ with cardinality less than that of $T$. We present necessary and sufficient conditions for the existence of a local domain $A$ with completion $T$, such that the generic formal fiber of $A$ has maximal elements equal to the ideals in $C$ and the generic formal fiber of $A[x_1,\ldots,x_n]_{(M\cap A,x_1,\ldots,x_n)}$ contains every element of $B_1$ but no element of $B_2$. If $T$ has characteristic $0$, we present necessary and sufficient conditions for the existence of an excellent local domain $A$ with the above properties.

We describe a closed immersion from each representation space of a type $A$ quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This ``bipartite Zelevinsky map'' restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type $A$ quivers of arbitrary orientation, we give the same result up to some factors of general linear groups.

These identifications allow us to recover results of Bobi\'nski and Zwara; namely, we see that orbit closures of type $A$ quivers are normal, Cohen-Macaulay and have rational singularities. We also see that each representation space of a type $A$ quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.