Computing in algebraic geometry and commutative algebra using Macaulay 2

AbstractWe present recent research of Eisenbud, Fløystad, Schreyer, and others, which was discovered with the help of experimentation with Macaulay 2. In this invited, expository paper, we start by considering the exterior algebra, and the computation of Gröbner bases. We then present, in an elementary manner, the explicit form of the Bernstein–Gelfand–Gelfand relationship between graded modules over the polynomial ring and complexes over the exterior algebra, that Eisenbud, Fløystad and Schreyer found. We present two applications of these techniques: cohomology of sheaves, and the construction of determinantal formulae for (powers of) Macaulay resultants. We show how to use Macaulay 2 to perform these computations

Solving multivariate polynomial systems and an invariant from commutative algebra

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Gr\"obner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate the solving degree of a system, focusing on systems arising within public-key cryptography. In particular, we show that it is upper bounded by, and often equal to, the Castelnuovo Mumford regularity of the ideal generated by the homogenization of the equations of the system, or by the equations themselves in case they are homogeneous. We discuss the underlying commutative algebra and clarify under which assumptions the commonly used results hold. In particular, we discuss the assumption of being in generic coordinates (often required for bounds obtained following this type of approach) and prove that systems that contain the field equations or their fake Weil descent are in generic coordinates. We also compare the notion of solving degree with that of degree of regularity, which is commonly used in the literature. We complement the paper with some examples of bounds obtained following the strategy that we describe

Cohomology for Frobenius kernels of SL2SL_2

Let $(SL_2)_r$ be the $r$-th Frobenius kernels of the group scheme $SL_2$ defined over an algebraically field of characteristic $p>2$. In this paper we give for $r\ge 1$ a complete description of the cohomology groups for $(SL_2)_r$. We also prove that the reduced cohomology ring \opH^\bullet((SL_2)_r,k)_{\red} is Cohen-Macaulay. Geometrically, we show for each $r\ge 1$ that the maximal ideal spectrum of the cohomology ring for $(SL_2)_r$ is homeomorphic to the fiber product G\times_B\fraku^r. Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobenius-Luzstig kernels of quantized enveloping algebras of type $SL_2$.Comment: published version; a section for the case p=2 is adde

Commuting varieties of rr-tuples over Lie algebras

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p$ and let \g be the Lie algebra of $G$. It is well known that for $p$ large enough the spectrum of the cohomology ring for the $r$-th Frobenius kernel of $G$ is homeomorphic to the commuting variety of $r$-tuples of elements in the nilpotent cone of \g [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \textbf{10} (1997), 693--728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties C_r(\mathfrak{gl}_2), C_r(\fraksl_2) and $C_r(\N)$ where $\N$ is the nilpotent cone of \fraksl_2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when \g=\fraksl_2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of \fraksl_3, we are able to verify the aforementioned properties for C_r(\fraku). Finally, applying our calculations on the commuting variety C_r(\overline{\calO_{\sub}}) where \overline{\calO_{\sub}} is the closure of the subregular orbit in \fraksl_3, we prove that the nilpotent commuting variety $C_r(\N)$ has singularities of codimension $\ge 2$.Comment: To appear in Journal of Pure and Applied Algebr

A commutative algebraic approach to the fitting problem

Given a finite set of points $\Gamma$ in $\mathbb P^{k-1}$ not all contained in a hyperplane, the "fitting problem" asks what is the maximum number $hyp(\Gamma)$ of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s). If $\Gamma$ has the property that any $k-1$ of its points span a hyperplane, then $hyp(\Gamma)=nil(I)+k-2$, where $nil(I)$ is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of $\Gamma$. Note that in $\mathbb P^2$ any two points span a line, and we find that the maximum number of collinear points of any given set of points $\Gamma\subset\mathbb P^2$ equals the index of nilpotency of the corresponding ideal, plus one.Comment: 8 page