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We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $ell$ give maximal rank for all the multiplication maps $times ell: [A]_i longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the emph{non-Lefschetz locus}. For monomial complete intersections, we completely describe the non-Lefschetz locus. For general complete intersections of codimension three and four we prove that the non-Lefschetz locus has the expected codimension, which in particular means that it is empty in a large family of examples. For general Gorenstein algebras of codimension three with a given Hilbert function, we prove that the non-Lefschetz locus has the expected codimension if the first difference of the Hilbert function is of decreasing type. For completeness we also give a full description of the non-Lefschetz locus for artinian algebras of codimension two.
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We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $mathbb P^2$ to $mathbb P^3$, and Hesse configurations in $mathbb P^2$.
Given an ideal $I=(f_1,ldots,f_r)$ in $mathbb C[x_1,ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $mathbb C[x_1,ldots,x_n]/I^k$ be? If $rle n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S/I^k$, where $I$ is the ideal generated by the $d$th powers of the variables.
Let $Z$ be a closed subscheme of a smooth complex projective variety $Ysubseteq Ps^N$, with $dim,Y=2r+1geq 3$. We describe the intermediate Neron-Severi group (i.e. the image of the cycle map $A_r(X)to H_{2r}(X;mathbb{Z})$) of a general smooth hypersurface $Xsubset Y$ of sufficiently large degree containing $Z$.
The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $mathbb P^3$ of degree $d geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d geq 4$, a very general surface in a simplicial toric threefold $mathbb P_Sigma$ (with orbifold singularities) has the same Picard number as $mathbb P_Sigma$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $mathbb P_Sigma$ in a linear system of a Cartier ample divisor with respect to a (-1)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve that is minimal in a suitable sense.
A finitely generated module over a commutative noetherian ring of finite Krull dimension can be built from the prime ideals in the singular locus by iteration of three procedures: taking extensions, direct summands, and cosyzygies. In 2003 Schoutens gave a bound on the number of iterations required to build any module, and in this note we determine the exact number. This building process yields a stratification of the module category, which we study in detail for local rings that have an isolated singularity.
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