16 research outputs found
On the tangent space to the Hilbert scheme of points in P3
No full textIn this paper we study the tangent space to the Hilbert scheme
, motivated by Haiman's work on and by a long-standing conjecture of Brian\c{c}on and Iarrobino
on the most singular point in . For points
parametrizing monomial subschemes, we consider a decomposition of the tangent
space into six distinguished subspaces, and show that a fat point exhibits an
extremal behavior in this respect. This decomposition is also used to
characterize smooth monomial points on the Hilbert scheme. We prove the first
Brian\c{c}on-Iarrobino conjecture up to a factor of 4/3, and improve the known
asymptotic bound on the dimension of .
Furthermore, we construct infinitely many counterexamples to the second
Brian\c{c}on-Iarrobino conjecture, and we also settle a weaker conjecture of
Sturmfels in the negative.Comment: 20 pages. Final version; to appear on Transactions of the American
Mathematical Societ
On Rees algebras of 2-determinantal ideals
No full textLet I be the ideal of minors of a 2 by n matrix of linear forms with the
expected codimension. In this paper we prove that the Rees algebra of I and its
special fiber ring are Cohen-Macaulay and Koszul; in particular, they are
quadratic algebras. The main novelty in our approach is the analysis of a
stratification of the Hilbert scheme of determinantal ideals. We study
degenerations of Rees algebras along this stratification, and combine it with
certain squarefree Groebner degenerations.Comment: Final version. To appear on the Journal of the London Mathematical
Societ
Hilbert schemes with two Borel-fixed points
No full textWe characterize Hilbert polynomials that give rise to Hilbert schemes with
two Borel-fixed points and determine when the associated Hilbert schemes or
their irreducible components are smooth. In particular, we show that the
Hilbert scheme is reduced and has at most two irreducible components. By
describing the singularities in a neighbourhood of the Borel-fixed points, we
prove that the irreducible components are Cohen-Macaulay and normal. We end by
giving many examples of Hilbert schemes with three Borel-fixed points.Comment: To appear in Journal of Algebr
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The Geometry of Hilbert Schemes on Projective Space
No full textIn this thesis we study singularities of Hilbert schemes and show that there are many (components) of Hilbert schemes that are smooth or mildly singular and use them to explore phenomena in birational geometry and commutative algebra. Specifically, we study the Hilbert scheme compactification of a pair of linear spaces, describe all the sub- schemes parameterized by this component and show that it is a smooth Mori dream space. We study Hilbert schemes with two Borel-fixed points and prove that they are reduced, and that their irreducible components have normal and Cohen-Macaulay singularities. We study the Hilbert scheme of points on a threefold and extend results on the Hilbert scheme of points of a surface to this case; we also provide bounds on the dimension of this Hilbert scheme. Finally, we generalize the Hilbert and Quot schemes to construct the fiber-full scheme, which is a fine moduli space that controls all the cohomological data of a variety instead of just the Hilbert polynomial
On the smoothness of lexicographic points on Hilbert schemes
No full textWe study the geometry of standard graded Hilbert schemes of polynomial rings
and exterior algebras. Our investigation is motivated by a famous theorem of
Reeves and Stillman for the Grothendieck Hilbert scheme, which states that the
lexicographic point is smooth. By contrast, we show that, in standard graded
Hilbert schemes of polynomial rings and exterior algebras, the lexicographic
point can be singular, and it can lie in multiple irreducible components. We
answer questions of Peeva-Stillman and of Maclagan-Smith.Comment: 13 pages. To appear on Journal of Pure and Applied Algebr
Rational singularities of nested Hilbert schemes
No full textThe Hilbert scheme of points of a smooth surface is
a well-studied parameter space, lying at the interface of algebraic geometry,
commutative algebra, representation theory, combinatorics, and mathematical
physics. The foundational result is a classical theorem of Fogarty, stating
that is a smooth variety of dimension . In recent
years there has been growing interest in a natural generalization of
, the nested Hilbert scheme ,
which parametrizes nested pairs of zero-dimensional subschemes of with . In contrast to Fogarty's theorem,
is almost always singular, and very little is
known about its singularities. In this paper we aim to advance the knowledge of
the geometry of these nested Hilbert schemes. Work by Fogarty in the 70's shows
that is a normal Cohen-Macaulay variety, and Song
more recently proved that it has rational singularities. In our main result, we
prove that the nested Hilbert scheme has rational
singularities. We employ an array of tools from commutative algebra to prove
this theorem. Using Gr\"obner bases, we establish a connection between
and a certain variety of matrices with an action of
the general linear group. This variety of matrices plays a central role in our
work, and we analyze it by various algebraic techniques, including square-free
Gr\"obner degenerations, the Stanley-Reisner correspondence, and the
Kempf-Lascoux-Weyman technique of calculating syzygies. Along the way, we also
obtain results on classes of irreducible and reducible nested Hilbert schemes,
dimension of singular loci, and -singularities in positive characteristic.Comment: 37 pages. Final version; to appear on International Mathematics
Research Notice
On the parity conjecture for Hilbert schemes of points on threefolds
No full textLet be the Hilbert scheme of points in , and let
denote the tangent space to a point . Okounkov and
Pandharipande have conjectured that and have the same parity for
every . For points parametrizing monomial ideals, the conjecture was
proved by Maulik, Nekrasov, Okounkov, and Pandharipande. In this paper, we
settle the conjecture for points parametrizing homogeneous ideals. In fact,
we state a generalization of the conjecture to Quot schemes of , and we
prove it for points parametrizing graded modules.Comment: Final version. To appear on The Annali della Scuola Normale Superiore
di Pisa, Classe di Scienz
A local study of the fiber-full scheme
No full textWe study some of the local properties of the fiber-full scheme, which is a
fine moduli space that generalizes the Hilbert scheme by parametrizing closed
subschemes with prescribed cohomological data. As a consequence, we provide
sufficient conditions for cohomology to remain constant under Gr\"obner
degenerations. We also describe a tangent-obstruction theory for the fiber-full
scheme in analogy with the one for the Hilbert scheme.Comment: to appear in Journal of Algebr
