olvido
This commit is contained in:
commit
4f16a82376
@ -5,7 +5,23 @@
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"execution_count": 1,
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"id": "72d5a30c-1a59-4a55-9680-fdfa7979625b",
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html>\\(\\displaystyle z^{3} - 3 y^{2} z + 3 x^{2} z + 2 x^{4}\\)</html>"
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],
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"text/latex": [
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"$\\displaystyle z^{3} - 3 y^{2} z + 3 x^{2} z + 2 x^{4}$"
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],
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"text/plain": [
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"z^3 - 3*y^2*z + 3*x^2*z + 2*x^4"
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]
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},
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||||
"metadata": {},
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||||
"output_type": "display_data"
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}
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],
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"source": [
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"# load('zetas_triples_general.py')\n",
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"R.<x, y, z> = PolynomialRing(QQ, order='neglex')\n",
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@ -13,7 +29,8 @@
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"f = v^2 - u^2\n",
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"g = u^4\n",
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"F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)\n",
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"h = F.discriminant(z)(x=u, y=v)"
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"h = F.discriminant(z)(x=u, y=v)\n",
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"show(F)"
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]
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},
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{
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@ -72,6 +89,21 @@
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"id": "27e92ddc-34d6-4290-880f-9d3561a8ec4e",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html>\\(\\displaystyle z^{3} - 3 y^{2} z + t x z^{2} + 3 x^{2} z + 2 x^{4}\\)</html>"
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],
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"text/latex": [
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"$\\displaystyle z^{3} - 3 y^{2} z + t x z^{2} + 3 x^{2} z + 2 x^{4}$"
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],
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"text/plain": [
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"z^3 - 3*y^2*z + t*x*z^2 + 3*x^2*z + 2*x^4"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/plain": [
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@ -85,12 +117,40 @@
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],
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"source": [
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"Ft = F + t * x * z^2\n",
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"show(Ft)\n",
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"Ft.jacobian_ideal().vector_space_dimension()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "83328b3a-8da8-4adc-8b6c-02764b1420d3",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html>\\(\\displaystyle 108 y^{6} + \\left(9 t^{2} - 324\\right) x^{2} y^{4} + \\left(-18 t^{2} + 324\\right) x^{4} y^{2} - 108 t x^{5} y^{2} + \\left(9 t^{2} - 108\\right) x^{6} + \\left(-8 t^{3} + 108 t\\right) x^{7} - 108 x^{8}\\)</html>"
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],
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"text/latex": [
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"$\\displaystyle 108 y^{6} + \\left(9 t^{2} - 324\\right) x^{2} y^{4} + \\left(-18 t^{2} + 324\\right) x^{4} y^{2} - 108 t x^{5} y^{2} + \\left(9 t^{2} - 108\\right) x^{6} + \\left(-8 t^{3} + 108 t\\right) x^{7} - 108 x^{8}$"
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],
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"text/plain": [
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"108*y^6 + (9*t^2 - 324)*x^2*y^4 + (-18*t^2 + 324)*x^4*y^2 + (-108*t)*x^5*y^2 + (9*t^2 - 108)*x^6 + (-8*t^3 + 108*t)*x^7 - 108*x^8"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"ht = Ft.discriminant(z)\n",
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"show(ht)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "8cc358aa-03d1-4b9c-b4ff-87b0324bd55b",
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"metadata": {},
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"outputs": [],
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@ -101,7 +161,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"execution_count": 7,
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"id": "d21a93b0-b0c4-4cd2-b648-f4bca8f78376",
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"metadata": {},
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"outputs": [],
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@ -112,7 +172,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"execution_count": 8,
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"id": "59240908-d672-4520-90e5-7358840f8c23",
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"metadata": {},
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"outputs": [
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@ -122,7 +182,7 @@
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"[(-108) * (-v + u)^3 * (v + u)^3]"
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]
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},
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"execution_count": 11,
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -133,7 +193,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"execution_count": 9,
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"id": "a1a4111c-050d-4656-be38-ceac87684b47",
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"metadata": {},
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"outputs": [
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@ -143,7 +203,7 @@
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"[(9) * (v - u)^2 * (v + u)^2 * (12*v^2 + (t^2 - 12)*u^2)]"
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]
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},
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"execution_count": 12,
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -154,7 +214,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"execution_count": 10,
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"id": "8ac2847d-121a-4a78-a202-7e1692998c99",
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"metadata": {},
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"outputs": [],
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@ -165,7 +225,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"execution_count": 11,
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"id": "e7884c6b-458b-4d7c-8c8f-376ec4b87c9a",
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"metadata": {},
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"outputs": [
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@ -175,7 +235,7 @@
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"[(108) * v^3 * (v + 2*u)^3, (108) * u^3 * (8*v^3 - u^5)]"
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]
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},
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"execution_count": 15,
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -186,7 +246,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"execution_count": 12,
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"id": "442b2c9c-c7f1-4fb9-ac12-bf372d90a342",
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"metadata": {},
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"outputs": [
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@ -197,7 +257,7 @@
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" ((4*t^2)) * u^4 * (9*v^2 + (-2*t)*u^3)]"
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]
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},
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"execution_count": 16,
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -260,6 +320,46 @@
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"[p.factor() for p in Ztb.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"id": "ea740cbb-5980-45e4-a6c9-695a78324c50",
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"metadata": {},
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"outputs": [],
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"source": [
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"Zt1 = ZetaFunctions(ht1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"id": "d98d3aec-353c-4dcf-9b8a-1ea5ca93512e",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[((-4*t^2)) * (-9*v^2 + (2*t)*u)]"
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]
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},
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"execution_count": 17,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"ht1 = S(ht(v=v + u)(v=u * v) / u^6)\n",
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"[p.factor() for p in Zt1.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "c78e9c40-ee08-4a4b-a0ca-4de12082d245",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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92
Mult3.tex
92
Mult3.tex
@ -401,7 +401,7 @@ The singular point is isolated if and only if the discriminant $D_zF$ is reduced
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The monic Weierstrass polynomial $z^3 -3b_2z + 2b_3$ is non degenerate unless $(c_2M_2)^3 = (c_3M_3)^2$.
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\end{proposition}
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\noindent \textcolor{blue}{Proof. The Newton polyhedron of $z^3 -3b_2z + 2b_3$ can be 2-dimensional, polygonal with two segments or polygonal with a unique segment. In the two first cases, it is non degenerate. The compact faces of the Newton polyhedron of $z^3 -3b_2z + 2b_3$ lie on a segment if and only if $(M_2)^3 = (M_3)^2$. The condition of $z^3 -3b_2z + 2b_3$ being non degenerate is equivalent to $z^3 -3b_2z + 2b_3 \in \CC[z]$ having a double root. Notice a triple root is not possible because the degree 2 monomial is zero. Assume that $z^3 -3b_2z + 2b_3 =(z-a)^2(z-b)$ for some $a, b \in \CC^\ast$. Equivalently, we have $2a+b=0$, and therefor $c_2=(\frac{b}{2})^2}$ and $c_3=(\frac{-b}{2})^3$.}
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\noindent \textcolor{blue}{Proof. The Newton polyhedron of $z^3 -3b_2z + 2b_3$ can be 2-dimensional, polygonal with two segments or polygonal with a unique segment. In the two first cases, it is non degenerate. The compact faces of the Newton polyhedron of $z^3 -3b_2z + 2b_3$ lie on a segment if and only if $(M_2)^3 = (M_3)^2$. The condition of $z^3 -3b_2z + 2b_3$ being non degenerate is equivalent to $z^3 -3b_2z + 2b_3 \in \CC[z]$ having a double root. Notice a triple root is not possible because the degree 2 monomial is zero. Assume that $z^3 -3b_2z + 2b_3 =(z-a)^2(z-b)$ for some $a, b \in \CC^\ast$. Equivalently, we have $2a+b=0$, and therefor $c_2=(\frac{b}{2})^2$ and $c_3=(\frac{-b}{2})^3$.}
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\section{Jung´s method for power series of multiplicity three }
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@ -445,7 +445,7 @@ $N^\Delta_j - 3 N^0_j = N^\Delta_j - 2 N^\infty_j$ is called the \emph{excess of
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\begin{example} Consider $F=z^3 - 3b_2z + 2b_3$ with
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$b_2= ((y^2 + x^3)^2 + x^5y)((x^2 + y^3)^2 + y^5x)$
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and
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$b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_3 D_zF$ is obtained after 30 blow-ups. The following figure and table give the dual resolution graph of $G$ and its associated numerical data.
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$b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_3 D_zF$ is obtained after 17 blow-ups. The following figure and table give the dual resolution graph of $G$ and its associated numerical data.
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@ -587,6 +587,9 @@ N^g_i& {\bf12}& {\bf 13}& {\bf 26}& {\bf 39}& {\bf 52}& {\bf 65}& {\bf 78}& {\bf
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N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& {\bf 52}&{\bf 104}\\
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\hline
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\hline
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\pi()& 1& 0& 0& 0& 0& 0& \mathbb{P}^1& 0&0\\
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\hline
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\hline
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\nu_i&2&3&5&7&9&11&13&6&11\\
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\chi()&-13&1&-1&0&0&0&-2&1&-1\\
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E_i^2& -11& -2& -4& -2& -2& -2& -1&-2&-1\\
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@ -676,7 +679,9 @@ in the case $I= \{i, j\}$.
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\begin{example} The following tables show all the trinomial forms and differential forms for the all the 0-dimensional strata and all the 1-dimensional strata with not zero Euler characteristic of Example \ref{extriplepole}
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$$\begin{array}{c|c|c|c}
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$$\begin{array}{c|c|c|c}
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i&f_i&\omega_i& \chi()\\
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\hline
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1& z^3 + x^4z^2 + x^{13}& (2, 1, 1)& -13\\
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@ -688,7 +693,6 @@ i&f_i&\omega_i& \chi()\\
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\end{array}$$
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$$\begin{array}{c|c|c}
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(i,j)&f_{i,j}&\omega_{i,j}\\
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\hline
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@ -1044,18 +1048,22 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
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\begin{proposition}\label{formestrpi}
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The monodromy zeta function $\zeta_F(t)$ and the local topological zeta function $Z(F,s)_0$ of $F$ equal
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$$(1 - t^3)^{\alpha} \prod_{I} \zeta_{f_I}(t)^{-\chi(E_I^\circ)}$$
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$$(1 - t^3)^{-\alpha} \prod_{I} \zeta_{f_I}(t)^{\chi(E_I^\circ)}$$
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$$\sum_I \chi(E^\circ_I) Z(f_I, \omega_I)_{o_I}$$
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where $I$ varies over the statification associated to the resolution $\pi$ of $b_2b_3D_zF$, the exponent $\alpha$ is the number of points of blow-ups needed to obtain $\pi$, and $o_I$ is a point in $E^\circ_I$.
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\end{proposition}
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\begin{proof} It follows from the the formula for $\zeta_f$ in terms of a partial resolution \cite[Theorem 1]{GSLM97} and the Stratum principle for topological zeta functions \cite[Stratum Principle 1.2]{ABCNLM02}. Each of the $\alpha$ factors of the form $(1-t^3)$ of $\zeta_f$ comes from the intersection of Milnor fiber of $F$ at $o$ and the line thru a center of blow-up of $\pi$. Each of these intersection sets consists of 3 points that are cyclically permuted by the monodromy.
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\begin{proof} It follows from the formulae for $\zeta_f$ in terms of a resolution \cite{AC}, in terms of a modification \cite[Theorem 1]{GSLM97}, and the Stratum principle for topological zeta functions \cite[Stratum Principle 1.2]{ABCNLM02}. Let us discuss the case of $\zeta_F$. The Milnor fiber $M_F$ intersects the $z$-axis in a set $A$ of three points that it is invariant under the action of the Milnor monodromy, i.e., the three points are cyclically permuted. Hence, we have $$\zeta_F = \zeta_F_{|_{M_F \setminus A}} \cdot \zeta_F_{|_A} = \zeta_F_{|_{M_F \setminus A}} (1 - t^3) .$$
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In order to consider compute the term $\zeta_F_{|_{M_F \setminus A}} $ we consider the modification $\tilde{X} : =X \times \CC \rightarrow \CC^{3}$. Denote the pull back of the Milnor fibre under $\tilde{\pi}$ by $\tilde{M}$. Assume that $\pi$ is a composition of $\alpha$ point blow-ups. Denote by $B$ the intersection of $\tilde{M}$ with the $\alpha$ vertical planes thru the exceptional divisors of $\pi$. The set $B$ is invariant under the pull back of the Milnor monodromy. Since the exceptional locus of $\pi$ is a tree of $\alpha$ exceptional divisors isomorphic to $\mathbb{P}^1$, its Euler characteristic is $\alpha + 1$. Therefore,
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$$\zeta_F= \zeta_{\pi^\ast F} = \zeta_{\pi^\ast F_{|_{\tilde{M} \setminus B}}} \zeta_{\pi^\ast F_{|_B}} = \zeta_{\pi^\ast F_{|_{\tilde{M} \setminus B}}} (1 - t^3)^{a+1}.$$
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Finally, we can express $\zeta_{\pi^\ast F} = \prod_{I} \zeta_{f_I}(t)^{\chi(E_I^\circ)}.$$
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% Each of the $\alpha$ factors of the form $(1-t^3)$ of $\zeta_f$ comes from the intersection of Milnor fiber of $F$ at $o$ and the line thru a center of blow-up of $\pi$. Each of these intersection sets consists of 3 points that are cyclically permuted by the monodromy.
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\end{proof}
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\begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=15$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} we have
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$$\zeta_F(t) = \frac{(1-t^3)^{15} \zeta^{}_{f_{2}} \zeta^{2}_{f_{4}} \zeta^{2}_{f_{5}} \zeta^{2}_{f_{6}} \zeta^{}_{f_{8}} \zeta^{}_{f_{10}} \zeta^{2}_{f_{12}} \zeta^{2}_{f_{13}} \zeta^{2}_{f_{14}} \zeta^{}_{f_{16}} }
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{\zeta^{13}_{f_{1}} \zeta^{}_{f_{3}} \zeta^{4}_{f_{7}} \zeta^{}_{f_{9}} \zeta^{}_{f_{11}} \zeta^{4}_{f_{15}} \zeta^{}_{f_{17}} }= \Phi^3_1 \Phi^{15}_2 \Phi_3 \Phi^2_6 \Phi^2_{13} \Phi^{15}_{26} \Phi^2_{78}$$
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\begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=17$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} and using the simmetry of the resolution graph and the fact that the monodromy zeta function coincides for the 16 points of intersection of two different exceptional divisor, we have
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$$\zeta_F(t) = \frac{\zeta^{2}_{f_{2}} \zeta^{2}_{f_{8}} \zeta^{16}_{f_{1,7}} \zeta^{13}_{f_{1,16}} \zeta^{2}_{f_{7,29}} \zeta^{2}_{f_{7,33}} \zeta^{2}_{f_{9,31}} }
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{(1-t^3)^{17} \zeta^{13}_{f_{1}} \zeta^{2}_{f_{3}} \zeta^{4}_{f_{7}} \zeta^{2}_{f_{9}} }= \Phi^3_1 \Phi^{15}_2 \Phi_3 \Phi^2_6 \Phi^2_{13} \Phi^{15}_{26} \Phi^2_{78}$$
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where $\Phi_i$ is $i$-th the cyclotomic polymonial, and $Z(F,s)_0$ equals
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$$-13Z(f_1,s)_0 + \sum_{i=2,8,10,16}Z(f_i,s)_0 - \sum_{i=3, 9,11,17} Z(f_i,s)_0 - 2 \sum_{i=7,15}Z(f_i,s)_0 + \sum_{(i,j) \in S} Z(f_{i,j},s)_0=$$
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$$\frac{-1/26s^3 + 2749/6084s^2 + 1969/4056s + 1/8}{(s+1)(s+1/2)^3}$$
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@ -1304,9 +1312,69 @@ $$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega
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\subsection{Third reduction: $\QQ$-resolution of the discriminant}
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$$\begin{array}{c|c|c|c|c}
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i&f_i&\omega_i& \chi()& \zeta_{f_i}\\
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\hline
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1& z^3 + x^4z^2 + x^{13}& (2, 1, 1)& -13& ( \Phi_2^2 \Phi_4 \Phi_6 \Phi_{12} \Phi_{26})^{-1} \\
|
||||
2,10& z^3 + x^{10}z + x^{13}& (3, 1, 1)&1& \Phi_1 (\Phi_{39})^{-1}\\
|
||||
3,11& z^3 + x^{20}z + x^{26}& (5, 1, 1)& -1& \Phi_1 ({\Phi_6 \Phi_{39} \Phi_{78})^{-1}\\
|
||||
7,15& z^3 + \color{red}{x^{52}z} + x^{78}& (13, 1, 1)& -2&( \Phi_1 \Phi_2^2 \textcolor{red}{\Phi_3} \Phi_6^2 \Phi_{13}^2 \Phi_{26}^2 \Phi_{39}^2 \Phi_{78}^2 )^{-1}\\
|
||||
8, 16& z^3 + x^{21}z + x^{26}& (6, 1, 1)& 1& \Phi_1 (\Phi_6 \Phi_{39} \Phi_{78})^{-1}\\
|
||||
9,17& z^3 + x^{42}z + x^{52}& (11, 1, 1)& -1& \Phi_1 (\Phi_6 \Phi_{12} \Phi_{39} \Phi_{78} \Phi_{156})^{-1}\\
|
||||
\end{array}$$
|
||||
|
||||
$$\begin{array}{c|c|c|c}
|
||||
(i,j)&f_{i,j}&\omega_{i,j}\\
|
||||
\hline
|
||||
&&\\
|
||||
(1, 7), (1, 15)& z^3 + x^4y^{26}z^2 + x^{13}y^{78}& (2, 13, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
|
||||
(2,3), (10, 11)& z^3 + x^{10}y^{20}z + x^{13}y^{26}& (3, 5, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
|
||||
(3, 9), (11, 17)& z^3 + x^{20}y^{42}z + x^{26}y^{52}& (5, 11, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
|
||||
(3, 4), (11,12)& z^3 + x^{20}y^{28}z + x^{26}y^{39}& (5, 7, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
|
||||
(8, 9), (16,17)& z^3 + x^{21}y^{42}z + x^{26}y^{52}& (6, 11, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
|
||||
(4, 5), (12,13)& z^3 + x^{28}y^{36}z + x^{39}y^{52}& (7, 9, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
|
||||
(5, 6), (13,14)& z^3 + x^{36}y^{44}z + x^{52}y^{65}& (9, 11, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
|
||||
(6, 7), (14,15)& z^3 + x^{44}y^{52}z + x^{65}y^{78}& (11, 13, 1)& \Phi_1 \textcolor{red}{\Phi_3}\\
|
||||
(1, i), i=16, \dots, 28& z^3 + x^4z^2 + x^{13}y& (2, 1, 1)& (\Phi_2 \Phi_4 \Phi_6 \Phi_{12})^{-1} \\
|
||||
(9, 31), (17, 32)& z^3 + x^{42}yz + x^{52}& (11, 1, 1)&\Phi_1 (\Phi_6 \Phi_{12} \Phi_{39} \Phi_{78} \Phi_{156})^{-1} \\
|
||||
(7, 33), (15, 34)& z^3 + x^{52}z + x^{78}y& (13, 1, 1)& (\Phi_1 \Phi_2^2 \textcolor{red}{\Phi_3} \Phi_6^2 \Phi_{13}^2 \Phi_{26}^2 \Phi_{39}^2 \Phi_{78}^2 )^{-1} \\
|
||||
(7, 29) (15, 30)& z^3 + x^{26}z^2 + x^{78}y& (13, 1, 1)& (\Phi_2 \Phi_6 \Phi_{13} \Phi_{26} \Phi_{39} \Phi_{78})^{-1} \\
|
||||
%(12, 'f')& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\
|
||||
%(16, 'g')& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\
|
||||
%(16, 'd')& z^3 + x^{26}z^2 + x^{78}y& (13, 1, 1)
|
||||
\end{array}$$
|
||||
|
||||
|
||||
$$\begin{align}
|
||||
\Phi_1& =& 17 - 13 + 8 - 16 - 2 &=& 18\\
|
||||
\Phi_2 &=& 26 + 8 -13 -2 -4 &=& 15\\
|
||||
\Phi_3 &=& -17 + 4 + 16 - 2 &= &1\\
|
||||
\Phi_4 &=& 13 -13 &=& 0\\
|
||||
\Phi_6 &=& 13 +2 + 8 -2 + 2 -13 -2-2-4 &=& 2\\
|
||||
\Phi_{12} &=& 13 + 2 -13 -2 &=& 0\\
|
||||
\Phi_{13} &= &8 -2 - 4& =& 2\\
|
||||
\Phi_{26} &= &13 + 8 -2 -4 & =& 15\\
|
||||
\Phi_{39} &= &-2 + 2 - 8 -2 + 2 + 2 + 2 + 4& =& 0 \\
|
||||
\Phi_{78} &= &2 + 8 -2 + 2 - 2 - 2 - 4& =& 2\\
|
||||
\Phi_{156} &= & 2 - 2& = &0
|
||||
\end{align}$$
|
||||
|
||||
|
||||
$$\Phi_1^{18}\Phi_2^{15}\Phi_3\Phi_6^2\Phi_{13}^2\Phi_{26}^{15}\Phi_{78}^2$$
|
||||
|
||||
|
||||
|
||||
\section{On generic projections}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
% In general, the $z$-axis is not a generic direction of projection for the embedded topological class of the surface given by the Weierstrass polynomial $F=z^3-2f(x,y)z+3g(x,y)$. Therefore the discriminant of $F$ with respect to $z$ given by $f^2+g^3$ is not generic inside the same equivalence class. Let us describe a situation where it is possible to describe a generic topological deformation $F_t=z^3-2f_t(x,y)z+3g_t(x,y)$ of $F$ whose discriminant $f_t^3+g_t^2$ is generic too.
|
||||
|
||||
|
||||
@ -1533,6 +1601,10 @@ and dual resolution graph.
|
||||
|
||||
\end{example}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\bibliographystyle{amsplain}
|
||||
\bibliography{biblio}
|
||||
|
||||
|
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Reference in New Issue
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