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Register a new userThe Scheme of Monogenic Generators and its Twists
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Given an extension of algebras $B/A$, when is $B$ generated by a single element $theta in B$ over $A$? We show there is a scheme $mathcal{M}_{B/A}$ parameterizing the choice of a generator $theta in B$, a moduli space of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples. A choice of a generator $theta$ is a point of the scheme $mathcal{M}_{B/A}$. This inspires a local-to-global study of monogeneity, piecing together monogenerators over points, completions, open sets, and so on. Local generators may not come from global ones, but they often glue to twisted monogenerators that we define. We show a number ring has class number one if and only if each twisted monogenerator is in fact a global generator $theta$. The moduli spaces of various twisted monogenerators are either a Proj or stack quotient of $mathcal{M}_{B/A}$ by natural symmetries. The various moduli spaces defined can be used to apply cohomological tools and other geometric methods for finding rational points to the classical problem of monogenic algebra extensions.
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Associated to a Mukai flop X ---> X is on the one hand a sequence of equivalences D(X) -> D(X), due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of D(X), due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlovs derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.
We prove that for any $mathbb{P}^n$-functor all the convolutions (double cones) of the three-term complex $FHR xrightarrow{psi} FR xrightarrow{tr} Id$ defining its $mathbb{P}$-twist are isomorphic. We also introduce a new notion of a non-split $mathbb{P}^n$-functor.
We compute the categorical entropy of autoequivalences given by P-twists, and show that these autoequivalences satisfy a Gromov-Yomdin type conjecture.
We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme $mathrm L_H$ by applying the constructive methods of Grobner bases, for any given term order. Indeed, this structure does not depend on the term order, since it can be defined as the scheme representing the functor of liftings of $H$. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed $mathrm L_H$ in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to find interesting topological properties, as for instance that $mathrm L_H$ is connected and that its locus of radical liftings is open. Moreover, we show that every ideal defining an arithmetically Cohen-Macaulay scheme of codimension two has a radical lifting, giving in particular an answer to an open question posed by L. G. Roberts in 1989.
The Hilbert scheme $mathbf{Hilb}_{p(t)}^{n}$ parametrizes closed subschemes and families of closed subschemes in the projective space $mathbb{P}^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we consider schemes over a field $k$ of characteristic zero and we present a new proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian $mathbf{Gr}_{p(r)}^{N(r)}$, where $N(r)= h^0 (mathcal{O}_{mathbb{P}^n}(r))$. Moreover, we exhibit explicit equations defining it in the Plucker coordinates of the Plucker embedding of $mathbf{Gr}_{p(r)}^{N(r)}$. Our proof of existence does not need some of the classical tools used in previous proofs, as flattening stratifications and Gotzmanns Persistence Theorem. The degree of our equations is $text{deg} p(t)+2$, lower than the degree of the equations given by Iarrobino and Kleiman in 1999 and also lower (except for the case of hypersurfaces) than the degree of those proved by Haiman and Sturmfels in 2004 after Bayers conjecture in 1982. The novelty of our approach mainly relies on the deeper attention to the intrinsic symmetries of the Hilbert scheme and on some results about Grassmannian based on the notion of extensors.
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