Mercer (Czechia)

An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four

We construct a homotopy through proper rational holomorphic maps from the unit ball in C^2 to the unit ball in C^4, joining (z,w) -> (z^3,sqrt(3)zw,w^3,0) to (z,w) -> (z,w,0,0). The construction answers the explicit target-four question in the AIM list on the Cauchy-Riemann equat...

An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four

We construct a homotopy through proper rational holomorphic maps from the unit ball in C^2 to the unit ball in C^4, joining (z,w) -> (z^3,sqrt(3)zw,w^3,0) to (z,w) -> (z,w,0,0). The construction answers the explicit target-four question in the AIM list on the Cauchy-Riemann equat...

Smooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields

For every prime p, integer n >= 2, and degree d >= n+1, we construct a geometrically smooth hypersurface X in projective n-space over F_p of degree d such that #X(F_p) is not congruent to 1 modulo p. The construction stays over the specified prime field in every characteristic an...

Smooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields

For every prime p, integer n >= 2, and degree d >= n+1, we construct a geometrically smooth hypersurface X in projective n-space over F_p of degree d such that #X(F_p) is not congruent to 1 modulo p. The construction stays over the specified prime field in every characteristic an...

A Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve

We give an explicit integral projective curve whose Hilbert scheme of eighteen points has a rational component of dimension seventeen. This answers affirmatively Problem 20 in the 2010 AIM workshop list Components of Hilbert Schemes, even with the curve required to be integral an...

Maximal Finite-Set Stabilizers in Thompson's Group T

Let D = Z[1/2]/Z be the dyadic circle and let T be Thompson's orientation-preserving circle group. We prove that, for every nonempty finite set A contained in D, its setwise stabilizer is a maximal proper subgroup of countably infinite index in T. If |A| = k, then Stab_T(A) is is...

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

We give a computer-assisted proof that the very-compressed locus with Hilbert function (1,4,10,10) is the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space over an algebraically closed field of characteristic...

Nonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information

Let X_1,...,X_m be a nonempty finite bounded selfadjoint tuple with finite joint nonmicrostates free Fisher information, and let Delta be the generator of the closed polynomial free-gradient form. We prove that its heat semigroup cannot converge uniformly to the identity in L2 on...

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

We give a computer-assisted proof that the very-compressed locus with Hilbert function (1,4,10,10) is the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space over an algebraically closed field of characteristic...

Nonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information

Let X_1,...,X_m be a nonempty finite bounded selfadjoint tuple with finite joint nonmicrostates free Fisher information, and let Delta be the generator of the closed polynomial free-gradient form. We prove that its heat semigroup cannot converge uniformly to the identity in L2 on...

Maximal Finite-Set Stabilizers in Thompson's Group T

Let D = Z[1/2]/Z be the dyadic circle and let T be Thompson's orientation-preserving circle group. We prove that, for every nonempty finite set A contained in D, its setwise stabilizer is a maximal proper subgroup of countably infinite index in T. If |A| = k, then Stab_T(A) is is...

A Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve

We give an explicit integral projective curve whose Hilbert scheme of eighteen points has a rational component of dimension seventeen. This answers affirmatively Problem 20 in the 2010 AIM workshop list Components of Hilbert Schemes, even with the curve required to be integral an...