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Register a new userThe interpolation problem for a set of three fat points in $mathbb{P}^1timesmathbb{P}^1$
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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.
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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.
Let $X$ be a set of $K$-rational points in $P^1 times P^1$ over a field $K$ of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, and let $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper we investigate the module of Kaehler differentials $Omega^1_{R_Y/K}$. We describe this bigraded $R_Y$-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $X$ is a complete intersection or an almost complete intersection in $P^1 times P^1$. Moreover, we introduce a Kaehler different for $Y$ and use it to characterize reduced fat point schemes in $P^1 times P^1$ having the Cayley-Bacharach property.
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This paper considers the moduli spaces (stacks) of parabolic bundles (parabolic logarithmic flat bundles with given spectrum, parabolic regular Higgs bundles) with rank 2 and degree 1 over $mathbb{P}^1$ with five marked points. The stratification structures on these moduli spaces (stacks) are investigated. We confirm Simpsons foliation conjecture of moduli space of parabolic logarithmic flat bundles for our case.
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