{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "rocky-semester",
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "using Oscar"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "single-afghanistan",
   "metadata": {},
   "source": [
    "# Worksheet 4: Deformations of IHS, ICIS, and beyond"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "animal-sight",
   "metadata": {},
   "source": [
    "## Versal deformations of isolated hypersurface singularities\n",
    "\n",
    "We can consider an isolated hypersurface singularity $f \\colon (\\mathbb C^n, 0) \\to (\\mathbb C,0)$ \n",
    "up to right- ($\\mathcal R$) or contact-equivalence ($\\mathcal K)$. As already discussed \n",
    "in the previous lectures, we can write down a semi universal unfolding/deformation \n",
    "by choosing a monomial basis for the Milnor- or the Tjurina algebra, respectively. \n",
    "\n",
    "Let us do this for the $A_3$-hypersurface singularity $f = x^4 - y\\cdot z$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "infinite-animation",
   "metadata": {},
   "outputs": [],
   "source": [
    "R, (x,y,z) = QQ[\"x\", \"y\", \"z\"]\n",
    "f = x^4 - y*z\n",
    "J = ideal(R, minors(jacobi_matrix(f), 1))\n",
    "\n",
    "# Let us first manually gather a monomial basis for the Milnor algebra\n",
    "o = negdegrevlex(gens(R)) # a local ordering on the variables of R\n",
    "leadJ = leading_ideal(J, ordering=o)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "younger-finding",
   "metadata": {},
   "source": [
    "Now we can manually gather those monomials in the monomial diagram which are not in the leading ideal \n",
    "of the jacobian ideal. In this case, we eventually find $1, x, x^2$.\n",
    "\n",
    "To automatize this, let us write a function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "heated-rental",
   "metadata": {},
   "outputs": [],
   "source": [
    "function kbase_local(I::MPolyIdeal)\n",
    "    R = base_ring(I)\n",
    "    \n",
    "    o = negdegrevlex(gens(R))\n",
    "    leadI = leading_ideal(I, ordering=o)\n",
    "    \n",
    "    mon_base = elem_type(R)[]\n",
    "    d = 0\n",
    "    new_mons = [x for x in Oscar.all_monomials(R, d) if !(x in leadI)]\n",
    "    while length(new_mons) > 0 \n",
    "        mon_base = vcat(mon_base, new_mons)\n",
    "        d = d+1\n",
    "        new_mons = [x for x in Oscar.all_monomials(R, d) if !(x in leadI)]\n",
    "    end\n",
    "    return mon_base\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "medieval-billion",
   "metadata": {},
   "source": [
    "Does it reproduce the same result? Let's check."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "equivalent-preparation",
   "metadata": {},
   "outputs": [],
   "source": [
    "kbase_local(J)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "robust-richmond",
   "metadata": {},
   "source": [
    "We can now write down a semi-universal unfolding of $f$. First, we need to introduce new \n",
    "variables, the deformation parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "honey-coating",
   "metadata": {},
   "outputs": [],
   "source": [
    "function semi_universal_unfolding_local(f::MPolyElem)\n",
    "    R = parent(f)\n",
    "    J = ideal(R, minors(jacobi_matrix(f), 1))\n",
    "    mon_base = kbase_local(J)\n",
    "    mu = length(mon_base) # The Milnor number\n",
    "    n = ngens(R)\n",
    "    \n",
    "    # set up the new ring with the deformation parameters.\n",
    "    Rext, v = PolynomialRing(coefficient_ring(R), vcat(symbols(R), [Symbol(\"u$i\") for i in 1:mu]))\n",
    "    u = gens(Rext)[n+1:end]\n",
    "    \n",
    "    # set up the usual map R → Rext\n",
    "    inc = hom(R, Rext, gens(Rext)[1:n])\n",
    "    F = inc(f)\n",
    "    F = F + sum([t*inc(x) for (t, x) in zip(u, mon_base)])\n",
    "    return (F, gens(Rext)[1:n], gens(Rext)[n+1:end])\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "multiple-being",
   "metadata": {},
   "source": [
    "Let us check that it gives the expected result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cardiac-pierce",
   "metadata": {},
   "outputs": [],
   "source": [
    "F, X, U = semi_universal_unfolding_local(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "anonymous-kentucky",
   "metadata": {},
   "source": [
    "## The discriminant of an IHS"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "olive-helping",
   "metadata": {},
   "source": [
    "Let $F(x; t) \\colon (\\mathbb C^n, 0) \\times (\\mathbb C^\\mu,0) \\to (\\mathbb C,0)$ be the semi-universal unfolding \n",
    "of an isolated hypersurface singularity $f = f_0 = F(x; 0)$. The *discriminant* is defined as the set of \n",
    "parameters $t \\in \\mathbb C^\\mu$ for which the fiber $X_t =  f^{-1}(\\{0\\})$ is singular. Can we compute that locus?\n",
    "\n",
    "First, we need a method to compute the \"relative Jacobian matrix\" where we take the derivatives only with respect to the variables, not the parameters. We can pass the variables as a second argument and simply add that method \n",
    "to the existing `jacobi_matrix` function. \n",
    "\n",
    "Note that julia distinguishes between different *methods* of the same \n",
    "*function* according to the *signature*, i.e. the number of arguments and their types."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "posted-skiing",
   "metadata": {},
   "outputs": [],
   "source": [
    "function Oscar.jacobi_matrix(f::MPolyElem, x::Vector{T}) where {T<:MPolyElem}\n",
    "    R = parent(f)\n",
    "    all(x->(x in gens(R)), x) || error(\"The given elements are not all ring variables\")\n",
    "    n = length(x)\n",
    "    A = zero(MatrixSpace(R, n, 1))\n",
    "    for i in 1:n\n",
    "      A[i, 1] = derivative(f, x[i])\n",
    "    end\n",
    "    return A\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "professional-pacific",
   "metadata": {},
   "source": [
    "Having the above method, we can manually compute the equation for the discriminant $\\Delta$ using elimination \n",
    "orderings for the projection to the parameter space, cf. yesterday's lectures."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "narrow-windsor",
   "metadata": {},
   "outputs": [],
   "source": [
    "RU = parent(F) # The ring of the ambient space of the deformation \n",
    "JF = ideal(RU, minors(jacobi_matrix(F, X), 1)) + ideal(RU, F) # The relative Tjurina ideal \n",
    "\n",
    "# We need an elemination ordering for the projection to the deformation base\n",
    "o = degrevlex(gens(RU)[1:3])*negdegrevlex(gens(RU)[4:5])\n",
    "\n",
    "# compute a standard basis w.r.t this elimination ordering\n",
    "SB = std_basis(JF, o)\n",
    "\n",
    "# collect only those elements which are contained in the subring of the base\n",
    "disc_eqns = [f for f in SB if all(v->!(divides(leading_monomial(f, o), v))[1], gens(RU)[1:3])]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dirty-arabic",
   "metadata": {},
   "source": [
    "The specialists among us may recognize that this is indeed the formula for the discriminant \n",
    "of a (depressed) polynomial of degree $4$. It is also known as the \"swallow tail\", a famous \n",
    "singularity. \n",
    "\n",
    "We can investigate the singular locus of $\\Delta$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "amino-journal",
   "metadata": {},
   "outputs": [],
   "source": [
    "dec = primary_decomposition(singular_slocus(ideal(RU, disc_eqns)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "beautiful-paintball",
   "metadata": {},
   "source": [
    "What a mess! But luckily, we already picked out some special points on $\\Delta$ for you; \n",
    "namely \n",
    "\n",
    " * $t_1 = (0,0,-1)$ on the regular locus of $\\Delta$; \n",
    " * $t_2 = (-3, 8, -6)$ on the second component above; \n",
    " * $t_3 = (1, 0, -2)$ on the first component. \n",
    " \n",
    "Let's check that these points really lay on them. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "secret-pocket",
   "metadata": {},
   "outputs": [],
   "source": [
    "delta = disc_eqns[1]\n",
    "t1 = [0,0,0, 0,0, -1]\n",
    "@show evaluate(delta, t1)\n",
    "@show all(g->iszero(evaluate(g, t1)), gens(dec[2][1]))\n",
    "@show all(g->iszero(evaluate(g, t1)), gens(dec[1][1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bronze-precipitation",
   "metadata": {},
   "outputs": [],
   "source": [
    "t2 = [0,0,0, -3, 8, -6]\n",
    "@show evaluate(delta, t2)\n",
    "@show all(g->iszero(evaluate(g, t2)), gens(dec[2][1]))\n",
    "@show all(g->iszero(evaluate(g, t2)), gens(dec[1][1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "assumed-stroke",
   "metadata": {},
   "outputs": [],
   "source": [
    "t3 = [0,0,0, 1, 0, -2]\n",
    "@show evaluate(delta, t3)\n",
    "@show all(g->iszero(evaluate(g, t3)), gens(dec[2][1]))\n",
    "@show all(g->iszero(evaluate(g, t3)), gens(dec[1][1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "wrapped-transsexual",
   "metadata": {},
   "source": [
    "Let us analyze the singularities in the fibers for either one of the three parameters. First $t_1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "pressed-gates",
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = evaluate(F, R.([x,y,z, 0, 0, -1]))\n",
    "radical(singular_slocus(ideal(R, f1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "chicken-profile",
   "metadata": {},
   "source": [
    "So $f_1 = F(x,y,z; t_1)$ has an isolated singularity at the origin. Which one is it?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "equal-diagnosis",
   "metadata": {},
   "outputs": [],
   "source": [
    "classify(f1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "combined-gardening",
   "metadata": {},
   "source": [
    "In fact, the whole regular stratum of $\\Delta$ is characterized by the fibers having one single $A_1$-singularity.\n",
    "\n",
    "Let us repeat the same with $t_2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "strange-georgia",
   "metadata": {},
   "outputs": [],
   "source": [
    "f2 = evaluate(F, R.([x,y,z, -3, 8, -6]))\n",
    "radical(singular_slocus(ideal(R, f2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acceptable-maximum",
   "metadata": {},
   "source": [
    "The `classify` command can only analyze singularities at the origin (because it is using local orderings in the background) and, hence, we first have to shift our function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "naval-attendance",
   "metadata": {},
   "outputs": [],
   "source": [
    "shift = hom(R, R, [x+1, y, z])\n",
    "classify(shift(f2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "criminal-tucson",
   "metadata": {},
   "source": [
    "So we see that on this stratum of $\\Delta$ the fibers have precisely one $A_2$ singularity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "large-tracy",
   "metadata": {},
   "outputs": [],
   "source": [
    "f3 = evaluate(F, R.([x,y,z, 1, 0, -2]))\n",
    "radical(singular_slocus(ideal(R, f3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "resistant-skill",
   "metadata": {},
   "source": [
    "Here, we have two singularities and therefore need two shifts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "split-reynolds",
   "metadata": {},
   "outputs": [],
   "source": [
    "shift1 = hom(R, R, [x+1, y, z])\n",
    "classify(shift1(f3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "defensive-southeast",
   "metadata": {},
   "outputs": [],
   "source": [
    "shift2 = hom(R, R, [x-1, y, z])\n",
    "classify(shift2(f3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "handled-daisy",
   "metadata": {},
   "source": [
    "We conclude that on the third stratum, the fibers have two singularities of type $A_1$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "minute-cookie",
   "metadata": {},
   "source": [
    "## Deformations of complete intersections\n",
    "\n",
    "We use the $S_5$-singularity from before."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "handed-headset",
   "metadata": {},
   "outputs": [],
   "source": [
    "R, (x,y,z) = QQ[\"x\", \"y\", \"z\"]\n",
    "I = ideal(R, [x*y, x^2 + y^2 + z^2])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "every-lodge",
   "metadata": {},
   "source": [
    "The normal module is defined as $N = \\mathrm{Hom}_R(I, R/I)$. OSCAR knows how to handle `Hom`-modules, \n",
    "but we need to convert all the participants into modules, first."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "strategic-worth",
   "metadata": {},
   "outputs": [],
   "source": [
    "F1 = FreeMod(R, 1) # A free module over R of rank 1\n",
    "Imod, _ = sub(F1, [g*F1[1] for g in gens(I)]) # the submodule of F1 generated by I\n",
    "RmodI, _ = quo(F1, Imod) # the quotient by that submodule\n",
    "N, vec_to_hom = hom(Imod, RmodI)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "threaded-brook",
   "metadata": {},
   "source": [
    "The last command is a bit subtle. It returns a pair `(N, vec_to_hom)` consisting of the actual \n",
    "module `N` and a map `vec_to_hom`. While often we can discard any extra maps returned by \n",
    "constructors, we will need this one in the following. \n",
    "\n",
    "What does `vec_to_hom` do? Well, `N` is constructed as a `Hom`-module. A priori, this is just another \n",
    "finitely generated module, but also every element $v \\in N$ can be interpreted as a module homomorphism \n",
    "$ I \\to R/I$. The returned map `vec_to_hom` allows us to do that interpretation on the computer side \n",
    "so that we really get an honest morphism of modules."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "residential-symphony",
   "metadata": {},
   "outputs": [],
   "source": [
    "v = N[1] # the first generator of N\n",
    "typeof(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "progressive-announcement",
   "metadata": {},
   "source": [
    "We can try to apply `v` to an element of `Imod` directly, but this will fail."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "assigned-intensity",
   "metadata": {},
   "outputs": [],
   "source": [
    "g = Imod[2] # the second generator of I\n",
    "v(g) # This will complain!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "conditional-death",
   "metadata": {},
   "source": [
    "Converting `v` into a homomorphism makes everything work."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "authorized-momentum",
   "metadata": {},
   "outputs": [],
   "source": [
    "phi = vec_to_hom(v)\n",
    "@show phi\n",
    "phi(Imod[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "hungry-survival",
   "metadata": {},
   "source": [
    "We proceed with the construction of the $T^1(S_5)$. We need to divide by the image of the jacobian matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "involved-commissioner",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = gens(I)\n",
    "Df = jacobi_matrix(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "executed-victor",
   "metadata": {},
   "source": [
    "Now we need the complicated map \n",
    "$\\theta \\colon R^n \\to N, \\quad (v_1,\\dots,v_n) \\mapsto \\left( \\sum_j a_j \\cdot f_j \\mapsto \\sum_{i,j} v_i \\frac{\\partial f_j}{\\partial x_i} a_j \\right)$. \n",
    "For this we need the `vec_to_hom` map from before and take *preimages* of elements along this map. \n",
    "Eventually, we arrive at the following, complicated expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "professional-toddler",
   "metadata": {},
   "outputs": [],
   "source": [
    "Fn = FreeMod(R, ngens(R))\n",
    "theta = hom(Fn, N, [preimage(vec_to_hom, hom(Imod, RmodI, [Df[i,j]*RmodI[1] for j in 1:ngens(Imod)])) for i in 1:ngens(R)])\n",
    "S, _ = image(theta)\n",
    "T1, _ = quo(N, S)\n",
    "kbase(T1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "refined-kingston",
   "metadata": {},
   "source": [
    "## Versal deformations of arbitrary isolated singularities\n",
    "\n",
    "Singular (and hence Oscar) knows how to compute the semi-universal deformation of an arbitrary \n",
    "singularity according to Grauert's theorem. Let us demonstrate this in Pinkham's example (cf. the \n",
    "first lecture on determinantal singularities by Prof. Maria Ruas)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "tough-banana",
   "metadata": {},
   "outputs": [],
   "source": [
    "R, x = PolynomialRing(QQ, \"x\" => 1:5)\n",
    "M1 = R[x[1] x[2] x[3] x[4]; x[2] x[3] x[4] x[5]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "referenced-bible",
   "metadata": {},
   "outputs": [],
   "source": [
    "I = ideal(R, minors(M1, 2))\n",
    "inc, p = versal_unfolding(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "impossible-favorite",
   "metadata": {},
   "source": [
    "Apart from the verbose Singular output during the computation, this returns us a pair of two maps: the inclusion of \n",
    "the central fiber `inc` into the family and the projection `p` down to the parameter space. \n",
    "The domains and codomains of these maps are 'space germs'."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "hispanic-interstate",
   "metadata": {},
   "outputs": [],
   "source": [
    "domain(inc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "tired-controversy",
   "metadata": {},
   "outputs": [],
   "source": [
    "domain(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "impressed-satisfaction",
   "metadata": {},
   "outputs": [],
   "source": [
    "codomain(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "biblical-lease",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.7.3",
   "language": "julia",
   "name": "julia-1.7"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}