The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}$
 that lifts to ℤ/p
 7 but not ℤ/p
 8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnev’s universality theorem.

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Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

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Classical Algebraic Geometry

We prove that there exists a positive α such that for any integer d≥3 and any topological types S1,…,Sn of plane curve singularities, satisfying

$$$$

https://kluedo.ub.uni-kl.de/files/757/no_series_18.pdf

Plane curves of minimal degree with prescribed singularities

Varieties with minimal secant degree and linear systems of maximal dimension on surfaces

We prove that r independent homogeneous polynomials of the same degree d become de- pendent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose (d 1)-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called the Weak Lefschetz Property) and a differential geometric notion, concerning varieties that satisfy certain Laplace equations. In the toric case, some relevant examples are classified, and as a byproduct we provide counterexamples to Ilardi's conjecture.

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Laplace equations and the weak lefschetz property

Motivated by the study of congruences of lines, we consider linear systems of skew-symmetric matrices of order six and constant rank four. We prove that, up to projective automorphisms, there exist exactly four types of such linear systems of dimension two, giving four families all of the same dimension.

On linear spaces of skew-symmetric matrices of constant rank

In this paper we study the Hilbert scheme of Palatini threefolds X in P 5 . We prove that such a scheme has an irreducible component containing X which is birational to the Grassmannian Gð3; � P 14 Þ and we determine the exceptional locus of the birational map.

https://www.emis.de/journals/AG/2-4/2_371.pdf

On the Hilbert scheme of Palatini threefolds

We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree d in $$n+1$$
 variables, for any $$d\ge 2$$
 and $$n\ge 2$$
 . We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if $$n=2$$
 (resp. $$n=2,3$$
 ) all range between the lower and upper bound is covered, while if $$n\ge 3$$
 (resp. $$n\ge 4$$
 ) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for $$n=2$$
 the Mumford–Takemoto stability of the syzygy bundle associated with smooth monomial Togliatti systems.

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The minimal number of generators of a Togliatti system

Let $X$ be an integral projective variety of codimension two, degree $d$ and dimension $r$ and $Y$ be its general hyperplane section. The problem of lifting generators of minimal degree $\sigma$ from the homogeneous ideal of $Y$ to the homogeneous ideal of $X$ is studied. A conjecture is given in terms of $d$, $r$ and $\sigma$; it is proved in the cases $r=1,2,3$. A description is given of linear systems on smooth plane curves whose dimension is almost maximal.

Emilia Mezzetti

Papers

Laplace equations and the weak lefschetz property

On linear spaces of skew-symmetric matrices of constant rank

On the Hilbert scheme of Palatini threefolds

The minimal number of generators of a Togliatti system

Differential-geometric methods for the lifting problem and linear systems on plane curves