Box-shaped matrices and the defining ideal of certain blowup surfaces

The equations defining blowup algebras of height three Gorenstein ideals

We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An i…

Curves generating extremal rays in blowups of weighted projective planes

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which th…

Symbolic blowup algebras and invariants associated to certain monomial curves in ${\mathbb P}^3$

In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$ in ${\mathbb P}^3$ parametrized by $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd(d,m)=1$. The defining ideal of these curves is a set-theoretic complete intersection. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal ${\mathfrak p}:={\mathfrak p}_{ { \mathcal C}(q,m) }$ defining the curve ${\mathcal C}(q,m)$ is computed. We also give a formula…

Koszul blowup algebras associated to three-dimensional Ferrers diagrams

We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees algebra and the toric ring. We show that the toric ring is a Koszul Cohen--Macaulay normal domain, while the Rees algebra is Koszul and the defining ideal is of fiber type.

超曲面的爆破

Blowups of hypersurfaces

本文针对非正则的 hypersurface rings，研究了其极大理想 Rees 环的定义理想，给出了极小生成集，并确定了该代数的 Cohen-Macaulayness 等不变量。

The Rees Algebra for Certain Monomial Curves

In this article, we find the equations defining the Rees algebra for certain Monomial Curves explicitly and use them to prove that the blowup scheme is not smooth. This proves a conjecture of Francia in affirmative, which says that a dimension one prime in a regular local ring is a complete intersection if it has a smooth blowup.