Consider the rational map $\varphi :{\mathbb{P}}_{\mathbf{k}}^{n-1}\frac{[f_0:\cdots :f_n]}{⇢}{\mathbb{P}}_{\mathbf{k}}^n$ defined by homogeneous polynomials $f_0,\dots ,f_n$ of the same degree $d$ in a polynomial ring $R=\mathbf{k}[x_1,\dots ,x_n]$ over a field $\mathbf{k}$. Suppose that $I=(f_0,\dots ,f_n)$ is a height two perfect ideal satisfying $\mu \left(I_p\right)\le dim{R}_p$ for $p\in Spec\left(R\right)\setminus V(x_1,\dots ,x_n)$. We study the equations defining the graph of $\varphi$ whose coordinate ring is the Rees algebra $R\left[\mathit{It}\right]$. We provide new methods to construct these equations using the work of Buchsbaum and Eisenbud. Furthermore, for certain classes of ideals satisfying the conditions above, our methods lead to explicit equations defining Rees algebras of the ideals in these classes. These classes of examples are interesting in that there are no known methods to compute the defining ideal of the Rees algebra of such ideals. Our new methods also give effective criteria to check that $\varphi$ is birational onto its image.

Using Bridgeland stability conditions, we give sufficient criteria for a stable vector bundle on a smooth complex projective surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the plane. As an application, we compute the cohomology of such bundles for curves that lie in the plane or on Hirzebruch surfaces.

In this paper we obtain a complete characterization of pseudo-collarable $n$-manifolds for $n\ge 6$. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as an extension of Siebenmann’s dissertation that can be applied to manifolds with nonstable fundamental group at infinity, our main theorem can also be viewed as an extension of the recent Gu–Guilbault characterization of completable $n$-manifolds in a manner that is applicable to manifolds whose fundamental group at infinity is not peripherally stable.

We find some extensions of the Kraft–Russell generic equivalence theorem, and we improve a result of Dubouloz and Kishimoto.

Lattice polytopes, which possess the integer decomposition property (IDP for short), turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytope and the stable set polytope of a graph, the following conditions are equivalent: (i) the Cayley sum is Gorenstein; (ii) the Minkowski sum is Gorenstein; and (iii) the graph is perfect.

In this paper we give an asymptotic bound of the cardinality of Zariski multiples of particular irreducible plane singular curves. These curves have only nodes and cusps as singularities and are obtained as branched curves of ramified covering of the plane by surfaces isogenous to a product of curves with group $(\mathbb{Z}/2\mathbb{Z}{)}^k$. The knowledge of the moduli space of these surfaces will enable us to produce Zariski multiplets whose number grows subexponentially in function of their degree.

We introduce and study a local combinatorial condition, called the $5/9$-condition, on a simplicial complex, implying Gromov hyperbolicity of its universal cover. We hereby give an application of another combinatorial condition, called $8$-location, introduced by Damian Osajda. Along the way we prove the minimal filling diagrams lemma for $5/9$-complexes.

We show that the knot type of the link of a real analytic map germ with isolated singularity $f:({\mathbb{R}}^2,0)\to ({\mathbb{R}}^4,0)$ is a complete invariant for $C^0$-$\mathcal{A}$-equivalence. Moreover, we also prove that isolated singularity implies finite $C^0$-determinacy, giving an explicit estimate for its degree. For the general case of real analytic map germs, $f:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^p,0)$ ($n\le p$), we use the Lojasiewicz exponent associated with Mond’s double point ideal $I^2\left(f\right)$ to obtain some criteria of Lipschitz and analytic regularity.

The probabilistic study of the value-distributions of zeta-functions is one of the modern topics in analytic number theory. In this paper, we study a certain probability measure related to the value-distribution of the Lerch zeta-function. We prove that it has a density function, and we can explicitly construct it. Moreover, we prove an asymptotic formula for the number of zeros of the Lerch zeta-function on the right side of the critical line, whose main term is associated with the density function.