Wednesday 7, December 2022

A m2 INVITATION TO MARKED FAMILIES

Deformation theory @ AGATES

the package

MarkedFamiliesWarsaw.m2 (click to download)

Available at

marked families the begininning of the story

$p(t) \in \mathbb{Q}[t]$ Hilbert polynomial $\mathbb{K}[\mathbf{x}] = \mathbb{K}[x_0,\ldots,x_n]$ coordinate ring of $\mathbb{P}^n$

Consider

  • $\prec$ graded term order on $\mathbb{K}[\mathbf{x}]$
  • $\mathcal{M}_{p(t)} = \left\{ J \subset \mathbb{K}[\mathbf{x}]\ \middle\vert\ \begin{array}{l} J\ \text{monomial ideal}\\ \text{HP}_{\mathbb{K}[\mathbf{x}]/J}(t) = p(t) \end{array}\right\}$
  • $\mathbf{St}_\prec(J) = \big\{I \subset \mathbb{K}[\mathbf{x}]\ \big\vert\ \text{in}_\prec(I) = J\big\}$

$$ \mathbf{Hilb}^{p(t)}(\mathbb{P}^n) = \bigsqcup_{J \in \mathcal{M}_{p(t)}} \mathbf{St}_{\prec}(J) $$

Notari, Spreafico. A stratification of Hilbert schemes by initial ideals and applications, Manuscripta Math. (2000)

marked families Borel-fixed ideals

marked families marked sets and marked bases

$J \subset \mathbb{K}[\mathbf{x}]$ quasi-stable ideal, $\mathcal{P}_J$ Pommaret basis, $\mathcal{N}(J) = \{\text{monomials}\ \mathbf{x}^\alpha \notin J\}$

definition A $J$-marked set is a set of polynomials $$ F = \left\{ f_{\alpha} := \mathbf{x}^\alpha + \sum_{\mathbf{x}^\beta \in \mathcal{N}(J)_{\vert \alpha\vert}} a_{\alpha,\beta}\, \mathbf{x}^\beta \ \middle\vert\ \mathbf{x}^{\alpha} \in \mathcal{P}_{J},\ a_{\alpha,\beta} \in \mathbb{K}\right\} $$

definition A $J$-marked set $F = \{f_\alpha\ \vert\ \mathbf{x}^\alpha \in \mathcal{P}_J\}$ is called $J$-marked basis if $$ \mathbb{K}[\mathbf{x}] = (F) \oplus \langle \mathcal{N}(J) \rangle\qquad\Longleftrightarrow\qquad \mathbb{K}[\mathbf{x}]/(F) = \langle \mathcal{N}(J)\rangle $$

Cioffi, Roggero. Flat families by strongly stable ideals and a generalization of Groebner bases, J. Symb. Comp. (2011)

marked families polynomial reduction

theorem Given a $J$-marked set $F = \left\{ f_{\alpha}\ \middle\vert\ \mathbf{x}^\alpha \in \mathcal{P}_J \right\}$, we can define a noetherian procedure called $\ast$-reduction $$ p \in \mathbb{K}[\mathbf{x}] \qquad \stackrel{F}{\longrightarrow}_{\ast} \qquad r \in \left\langle\mathcal{N}(J)\right\rangle $$

theorem $F = \left\{ f_{\alpha}\ \middle\vert\ \mathbf{x}^\alpha \in \mathcal{P}_J \right\}$ $J$-marked set $$ F\ \text{is a}\ J\text{-marked basis} \qquad \Longleftrightarrow\qquad x_i f_{\alpha} \stackrel{F}{\longrightarrow}_{\ast} 0,\quad \forall\ x_i,\ \forall\ f_{\alpha} $$

functors locally closed

definition $J\subset \mathbb{K}[\mathbf{x}]$ quasi-stable ideal. Define the functor $$ \underline{\mathbf{Mf}}_J : (\mathbb{K}\text{-algebras}) \to (\text{Sets}) $$ such that $$ \begin{split} \underline{\mathbf{Mf}}_J(A) &{}= \left\{ I \subset A[\mathbf{x}]\ \middle\vert\ A[\mathbf{x}] = I \oplus \langle\mathcal{N}(J)\rangle\right\}\\ &{}= \left\{ I \subset A[\mathbf{x}]\ \middle\vert\ I = (F),\ J\text{-marked basis}\ F\right\} \end{split} $$

theorem $J \in \mathcal{M}_{p(t)}^{q.s.}$

  • $\underline{\mathbf{Mf}}_J$ is representable
  • $\mathbf{Mf}_J$ locally closed in $\mathbf{Hilb}^{p(t)}(\mathbb{P}^n)$

L., Roggero. On the functoriality of marked families, J. Commut. Algebra (2016)

functors open

$$ \begin{array}{c c cccc ccccc cc} \underline{\mathbf{Mf}}_J(A) & \subseteq & \underline{\mathbf{Mf}}_{J_{\geqslant 2}}(A) & \subseteq & & \ldots && \subseteq & \underline{\mathbf{Mf}}_{J_{\geqslant t}}(A) & \subseteq && \ldots \\ I & \mapsto & I_{\geqslant 2} & \mapsto && \ldots && \mapsto & I_{\geqslant t} & \mapsto && \ldots \\ \\ \end{array} $$

theorem For $t \gg 0$, $\underline{\mathbf{Mf}}_{J_{\geqslant t}}$ is an open subfunctor of the Hilbert functor, i.e. $$ \mathbf{Mf}_{J_{\geqslant t}} \underset{\text{open}}{\subset} \mathbf{Hilb}^{p(t)}(\mathbb{P}^n) $$

main features marked families

  • Usage
    • markedFamily J
  • Input
    • J quasi-stable ideal
  • Output
    • provides the generic $J$-marked set and the equations of $\mathbf{Mf}_J$

main features marked families

  • Usage
    • openSubsetHilbertScheme J
  • Input
    • J quasi-stable ideal
  • Output
    • provides the generic $J_{\geqslant t}$-marked set and the equations of the open subset $ \mathbf{Mf}_{J_{\geqslant t}}$

example 1 $\mathbf{Hilb}^{3m+1}(\mathbb{P}^3)$

example 1 $\mathbf{Hilb}^{3m+1}(\mathbb{P}^3)$

example 1 $\mathbf{Hilb}^{3m+1}(\mathbb{P}^3)$

main features marked families

  • Usage
    • markedFamily J
  • Input
    • J quasi-stable ideal
  • Output
    • provides the generic $J$-marked set and the equations of $\mathbf{Mf}_J$
  • Optional inputs
    • Minimize => ..., a Boolean value, default value false, whether to try to reduce the number of parameters

example 1 $\mathbf{Hilb}^{3m+1}(\mathbb{P}^3)$

main features tangent spaces

  • Usage
    • tangentSpaceMarkedFamily J
    • tangentSpaceMarkedFamily (J,I)
  • Input
    • J quasi-stable ideal
    • I homogeneous ideal
  • Output
    • provides the tangent space $T_{[J]}\mathbf{Mf}_J$
    • provides the tangent space $T_{[I]}\mathbf{Mf}_J$ if $I$ is generated by a $J$-marked basis

main features tangent spaces

  • Usage
    • tangentSpaceHilbertScheme J
    • tangentSpaceHilbertScheme (J,I)
    • tangentSpaceHilbertScheme I
  • Input
    • J quasi-stable ideal
    • I homogeneous ideal
  • Output
    • provides the tangent space $T_{[J]} \mathbf{Hilb}^{p(t)}(\mathbb{P}^n)\simeq T_{[J]}\mathbf{Mf}_{J_{\geqslant t}}$
    • provides the tangent space $T_{[I]} \mathbf{Hilb}^{p(t)}(\mathbb{P}^n)\simeq T_{[I]}\mathbf{Mf}_{J_{\geqslant t}}$, if $I$ is generated by a $J$-marked basis
    • provides the tangent space $T_{[I]} \mathbf{Hilb}^{p(t)}(\mathbb{P}^n)\simeq T_{[I]}\mathbf{Mf}_{J_{\geqslant t}}$, for a suitable quasi-stable ideal $J$

example 2 smoothability (1,5,5,1)

Given

  • $F \in \mathbb{K}[x_1,x_2,x_3,x_4,x_5]_3$
  • apolar ideal $I_F \in \mathbb{K}[a_1,a_2,a_3,a_4,a_5]$ with Hilbert function $(1,5,5,1)$

Show

  • $\dim T_{[I_F]} \mathbf{Hilb}^{12}(\mathbb{A}^5) = 12\cdot 5 =60$
  • $I_F$ has a smoothable deformation

Jelisiejew. Local finite-dimensional Gorenstein $k$-algebras having Hilbert function (1,5,5,1) are smoothable, J. Algebra Appl. (2014)

example 2 smoothability (1,5,5,1)

example 3 negative tangents

Consider $$ L \subset (x,y,z)^7 \subset R = \mathbb{K}[x,y,z] $$ such that

  • $[L] \in \mathbf{Hilb}^{96}(\mathbb{A}^3)$
  • Hilbert function $(1,3,6,10,15,21,28,12)$

Check

  • $\text{Hom}(L,R/L)_{< -1} = 0$

Compute

  • $\text{Hom}(L,R/L)_{= -1} = $ ??

example 3 negative tangents

example 4 unirationality of $\mathcal{M}_g$

For $\mathcal{M}_g,\ g \leqslant 10$, the unirationality proof uses plane models with double points in general position [Severi]

$$r=2 \qquad d = \min\{d\ \vert\ \rho(g,d,r) \geqslant 0\} \qquad \delta = \binom{d-1}{2}-g$$

Schreyer. Computer aided unirationality proofs of moduli spaces, Handbook of moduli. Vol. III (2013)

example 4 unirationality of $\mathcal{M}_g$

example 5 singular nested Hilbert schemes

$(J_1,J_2)$ pair of quasi-stable ideals s.t. $J_2 \subset J_1$: $$ \begin{split} & F_1 = \left\{ f_{\alpha}\ \middle\vert\ \mathbf{x}^\alpha \in \mathcal{P}_{J_1}\right\}\ \text{describes}\ \mathbf{Mf}_{J_1}\\ & F_2 = \left\{ g_{\beta}\ \middle\vert\ \mathbf{x}^\beta \in \mathcal{P}_{J_2}\right\}\ \text{describes}\ \mathbf{Mf}_{J_2} \end{split} $$

$H \subset \mathbf{Mf}_{J_1} \times \mathbf{Mf}_{J_2}$ open in $\mathbf{Hilb}^{(n_1,n_2)}(\mathbb{A}^3)$ defined by imposing $$ g_{\beta} \stackrel{F_1}{\longrightarrow}_\star 0,\qquad\forall\ g_\beta \in F_2 $$

example 5 singular nested Hilbert schemes

main features Quot schemes

$$ \begin{array}{ccc} \text{quasi-stable ideal}\ J& \qquad\leadsto\qquad & \text{quasi-stable submodule}\ \oplus J^{(k)} \mathbf{e}_k\\ \mathbf{Hilb}^{p(t)}(\mathbb{P}^n)& \qquad\leadsto\qquad & \mathbf{Quot}^{p(t)}(\mathcal{O}_{\mathbb{P}^n}^r)\\ \end{array} $$

  • Usage
    • markedFamily U openSubsetQuotScheme U
      tangentSpaceMarkedFamily U dimTangentSpaceMarkedFamily U
      tangentSpaceQuotScheme U dimTangentSpaceQuotScheme U
  • Input
    • U quasi-stable submodule
  • Output
    • ...

Albert, Bertone, Roggero, Seiler. Computing Quot schemes via marked bases over quasi-stable modules, J. Algebra (2020)

example 6 tangent space quot scheme

$$R = \mathbb{K}[x_0,x_1,x_2,x_3,x_4] \qquad\qquad\qquad M : R^{12}(-1) \to R^4$$ $$ \small\left[ \begin{array}{cccccccccccc} x_{0}&x_{1}&x_{2}&x_{3}&0&0&0&0&0&0&0&0\\ 0&0&0&0&x_{0}&x_{1}&x_{2}&x_{3}&0&0&0&0\\ 0&0&0&0&0&0&0&0&x_{0}&x_{1}&x_{2}&x_{3}\\ 0&0&x_{3}&x_{0}&x_{1}&0&0&2\,x_{0}+3\,x_{1}+x_{2}+2\,x_{3}&x_{1}+x_{2}&2\,x_{0}+3\,x_{1}+5\,x_{2}+5\,x_{3}&0&0 \end{array}\right] $$ $$ X = [\text{coker}(M)] \in \mathbf{Quot}^8(\mathcal{O}_{\mathbb{P}^4}^4)\qquad\qquad \dim T_X \mathbf{Quot}^8(\mathcal{O}_{\mathbb{P}^4}^4) = ?? $$

Jelisiejew, Šivic. Components and singularities of Quot schemes and varieties of commuting matrices, Crelles Journal (2022)

example 6 tangent space quot scheme