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$V$ is a complete intersection scheme in a multiprojective space if it can be defined by an ideal $I$ with as many generators as $textrm{codim}(V)$. We investigate the multigraded regularity of complete intersections scheme in $mathbb{P}^ntimes mathbb{P}^m$. We explicitly compute many values of the Hilbert functions of $0$-dimensional complete intersections. We show that these values only depend upon $n,m$, and the bidegrees of the generators of $I$. As a result, we provide a sharp upper bound for the multigraded regularity of $0$-dimensional complete intersections.
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We define logarithmic tangent sheaves associated with complete intersections in connection with Jacobian syzygies and distributions. We analyse the notions of local freeness, freeness and stability of these sheaves. We carry out a complete study of logarithmic sheaves associated with pencils of quadrics and compute their projective dimension from the classical invariants such as the Segre symbol and new invariants (splitting type and degree vector) designed for the classification of irregular pencils. This leads to a complete classification of free (equivalently, locally free) pencils of quadrics. Finally we produce examples of locally free, non free pencils of surfaces in P3 of any degree k at least 3, answering (in the negative) a question of Calvo-Andrade, Cerveau, Giraldo and Lins Neto about codimension foliations on P3 .
This paper is devoted to the study of multigraded algebras and multigraded linear series. For an $mathbb{N}^s$-graded algebra $A$, we define and study its volume function $F_A:mathbb{N}_+^sto mathbb{R}$, which computes the asymptotics of the Hilbert function of $A$. We relate the volume function $F_A$ to the volume of the fibers of the global Newton-Okounkov body $Delta(A)$ of $A$. Unlike the classical case of standard multigraded algebras, the volume function $F_A$ is not a polynomial in general. However, in the case when the algebra $A$ has a decomposable grading, we show that the volume function $F_A$ is a polynomial with non-negative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.
Let $(A,mathfrak{m})$ be an abstract complete intersection and let $P$ be a prime ideal of $A$. In [1] Avramov proved that $A_P$ is an abstract complete intersection. In this paper we give an elementary proof of this result.
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 $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(bf b)$ for a degree $mathbf{b}in A$. We show that the homology of $D_{gcd}(bf b)$ determines the $i$-Betti numbers of degree $bf b$. We discuss the notion of an indispensable complex of $R/I$. We show that the Koszul complex of a complete intersection lattice ideal $I$ is the indispensable resolution of $R/I$ when the $A$-degrees of the elements of the generating $R$-sequence are incomparable.
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