Related papers: Syzygies of Jacobian ideals and defects of linear …

Let $V$ be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for $V$ are exactly the…

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the…

Given a standard graded polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic zero and a graded $k$-subalgebra $A=k[f_1,...,f_m]\subset R$, one relates the module $\Omega_{A/k}$ of K\"ahler $k$-differentials of $A$ to the…

In this paper we give a characterization of strongly Euler homogeneous singular points on a reduced complex projective hypersurface $D=V(f)\subset \PP^n$ using the Jacobian syzygies of $f$. The characterization compares the ranks of the…

The Jacobian ideal provides the set of infinitesimally trivial deformations for a homogeneous polynomial, or for the corresponding complex projective hypersurface. In this article, we investigate whether the associated linear deformation is…

Let $\mathcal{A}$ denote a central hyperplane arrangement of rank $n$ in affine space $\mathbb{K}^n$ over an infinite field $\mathbb{K}$ and let $l_1,\ldots, l_m\in R:= \mathbb K[x_1,\ldots,x_n]$ denote the linear forms defining the…

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show…

Let $\mathcal C :f=0$ be a curve arrangement in the complex projective plane. If $\mathcal C$ contains a curve subarrangement consisting of at least three members in a pencil, then one obtains an explicit syzygy among the partial…

In this paper we derive geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics. These consequences…

Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There…

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of…

Let $V:f=0$ be a hypersurface of degree $d \geq 3$ in the complex projective space $\mathbb{P}^n$, $n \geq 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H: \ell=0$ be a hyperplane in…

We study the syzygies of a codimension two ideal I = <f_1,f_2,f_3> in k[x,y,z]. Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus Z = V(I) is generated by the Koszul syzygies iff Z is a local…

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface. The result gives information on the position of the singularities of a nodal hypersurface…

In this article, we consider the singularity of an arbitrary homogeneous polynomial with complex coefficients $f(x_0,\dots,x_n)$ at the origin of $\mathbb C^{n+1}$, via the study of the monodromy characteristic polynomials $\Delta_l(t)$,…

Let f_1,...,f_r be homogeneous polynomials in K[x_1,...,x_n], K a field. Put F=y_1f_1+...+y_rf_r in K[x,y] and let I be the ideal of K[x,y] generated by the partials of F relative to the x_i and y_j. The Jacobian ring of F is the quotient…

Let $I\subset \mathbb C[x,y,z]$ be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If $I$ is a locally complete intersection we give a criterion for $\mathbb C[x,y,z]/I$ to be arithmetically…

In this paper, we will first show that, the homogeneous polynomials which satisfy the Jacobian condition are injective on the lines that pass through the origin. Secondly, we will show that $F$ and $G'$ are paired, where $F$ is a Druzkowski…

To a generic hypersurface in the affine torus $(\mathbb{C}^*)^n$ we associate a hypersurface arrangement in the projective space $\mathbb{P}^n$ consisting of the $n+1$ coordinate hyperplanes and a generic hypersurface, and compute the…