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Register a new userSpecial arrangements of lines: codimension two ACM varieties in $mathbb P^1timesmathbb P^1timesmathbb P^1$
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In this paper we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimensional two arithmetically Cohen-Macaulay (ACM) varieties in $mathbb P^1timesmathbb P^1timesmathbb P^1$, called varieties of lines. We also describe their ACM property from combinatorial algebra point of view.
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This is an appendix to the recent paper of Favacchio and Guardo. In these notes we describe explicitly a minimal bigraded free resolution and the bigraded Hilbert function of a set of 3 fat points whose support is an almost complete intersection (ACI) in $mathbb{P}^1timesmathbb{P}^1.$ This solve the interpolation problem for three points with an ACI support.
We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement $mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.
A projectively normal Calabi-Yau threefold $X subseteq mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $mathrm{codim}(I)=4=mathrm{reg}(I)$, and that exactly 8 of these occur for smooth irreducible nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of $X$ with $h^{p,q}(X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties. A key tool in our approach is the use of inverse systems to identify possible betti tables for $X$.
We propose a conjectural explicit formula of generating series of a new type for Gromov--Witten invariants of $mathbb{P}^1$ of all degrees in full genera.
We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring $mathbb{F}_p[G]$ of an elementary abelian $p$-group $G$ in terms of commutative algebra. This extends results of Carlsson for $p=2$ to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over $mathbb{F}_p[G]$ into the derived category of $p$-DG modules over a polynomial ring.
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