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Alvaro Liendo

Publications and source records attributed to Alvaro Liendo.

At least 19 recordsLinked to original sources
arXiv2026

Twisted forms of classical hypersurfaces

We count the twisted forms, over the field of real numbers and over finite fields, of the three classical families of smooth hypersurfaces with large automorphism group: the Fermat, Delsarte and Klein hypersurfaces. Our main tool is a counting formula for the Galois cohomology set of a smooth hypersurface whose automorphism group is the semidirect product of a diagonal abelian group and a group of permutations of the monomials of its defining equation. In order to apply this formula over finite fields, we extend the differential method of Oguiso and Yu to positive characteristic, and we compute the automorphism groups of the Fermat, Delsarte and Klein hypersurfaces over algebraically closed fields of positive characteristic under explicit arithmetic conditions on p. Over the reals, our count recovers a recent theorem of Sasaki on the real forms of Fermat hypersurfaces.

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arXiv2026

Automorphism groups of rigid complete intersections

We study the automorphism groups of complete intersections of hypersurfaces of strictly increasing degrees in projective space. Under a combinatorial rigidity condition on the tuple of defining polynomials, we show that every automorphism of the complete intersection extends to an automorphism of each defining hypersurface, so that its automorphism group is the intersection of the automorphism groups of the defining hypersurfaces. We apply this principle to two natural families of complete intersections of two hypersurfaces of different degrees. For complete intersections of two Fermat hypersurfaces, we determine the automorphism group in every smooth case. For complete intersections of a Klein hypersurface with the reverse-order Klein hypersurface, we describe the automorphism group under an explicit arithmetic condition relating the two degrees, with Klein hypersurfaces of Wagstaff type as a natural source of examples.

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arXiv2026

A one-step counterexample to the normalized Nash blowup conjecture

We construct an explicit normal singular affine toric variety X of dimension five over an algebraically closed field of characteristic three such that the normalized Nash blowup of X already contains an open affine subset isomorphic to X. Combined with previously known examples, this yields one-step counterexamples in every dimension greater than or equal to five and every characteristic. The characteristic-three case is the most delicate: the previously known counterexample in dimension four requires a two-step iteration of the normalized Nash blowup, and our example demonstrates that in dimension five and higher the minimal number of iterations needed to produce a loop is one.

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arXiv2025

On a Computational Approach to the Nash Blowup Problem

In this paper we describe the implementation that led to the counterexamples to the Nash blowup conjectures recently discovered by the authors. We also provide new examples of toric varieties with prescribed singularities that are not resolved by the normalized Nash blowup, including cyclic quotient singularities, toric hypersurfaces, and Q-factorial Gorenstein singularities. In addition, we report extensive computational evidence: tens of thousands of two-dimensional toric varieties that are resolved by iterating the Nash blowup, and millions of three-dimensional toric varieties that are resolved by iterating the normalized Nash blowup. This provides positive evidence for the remaining open cases of the conjectures.

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arXiv2025

Automorphisms of prime power order of weighted hypersurfaces

We study automorphisms of quasi-smooth hypersurfaces in weighted projective spaces, extending classical results for smooth hypersurfaces in projective space to the weighted setting. We establish effective criteria for when a power of a prime number can occur as the order of an automorphism, and we derive explicit bounds on the possible prime orders. A key role is played by a weighted analogue of the classical Klein hypersurface, which we show realizes the maximal prime order of an automorphism under suitable arithmetic conditions. Our results generalize earlier work by Gonz\'alez-Aguilera and Liendo.

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arXiv2025

Characteristic-free normalized Nash blowup of toric varieties

We introduce conditions on cones of normal toric varieties under which the polyhedron defining the normalized Nash blowup does not depend on the characteristic of the base field. As a consequence, we deduce several results on the resolution of singularities properties of normalized Nash blowups. In particular, we recover all known results of the families that can be resolved via normalized Nash blowups in positive characteristic. We also provide new families of toric varieties whose normalized Nash blowup is non-singular in arbitrary characteristic.

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arXiv2024

Codimension two torus actions on the affine space

In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface. These varieties are of particular interest as they represent the simplest candidates for potential counterexamples to the linearization conjecture in affine geometry.

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arXiv2024

The Automorphism groups of zero-dimensional monomial algebras

A monomial algebra B is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra B, under the assumption that B is a finite-dimensional vector space over a field of characteristic zero. We achieve this by providing an explicit classification of the homogeneous locally nilpotent derivations of B. The main body of the paper addresses the more general case of semigroup algebras, with the polynomial ring being a particular case.

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arXiv2024

On the characterization of affine toric varieties by their automorphism group

In this paper we show that a normal affine toric variety X different from the algebraic torus is uniquely determined by its automorphism group in the category of affine irreducible, not necessarily normal, algebraic varieties if and only if X is isomorphic to the product of the affine line and another affine toric variety. In the case where X is the algebraic torus T, we reach the same conclusion if we restrict the category to only include irreducible varieties of dimension at most the dimension of T. There are examples of varieties of dimension one higher than T having the same automorphism group of T. Hence, this last result is optimal.

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arXiv2024

On a Torelli Principle for automorphisms of Klein hypersurfaces

Using a refinement of the differential method introduced by Oguiso and Yu, we provide effective conditions under which the automorphisms of a smooth degree $d$ hypersurface of $\mathbf{P}^{n+1}$ are given by generalized triangular matrices. Applying this criterion we compute all the remaining automorphism groups of Klein hypersurfaces of dimension $n\geq 1$ and degree $d\geq 3$ with $(n,d)\neq (2,4)$. We introduce the concept of extremal polarized Hodge structures, which are structures that admit an automorphism of large prime order. Using this notion, we compute the automorphism group of the polarized Hodge structure of certain Klein hypersurfaces that we call of Wagstaff type, which are characterized by the existence of an automorphism of large prime order. For cubic hypersurfaces and some other values of $(n,d)$, we show that both groups coincide (up to involution) as predicted by the Torelli Principle.

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arXiv2023

On the automorphism group of non-necessarily normal affine toric varieties

Our main result is the following: let X be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface X' with automorphism group isomorphic to the automorphism group of X if and only if X is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety X of any dimension. Recall that normalized additive group actions on X are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of X. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.

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arXiv2022

On rational multiplicative group actions

We establish a one-to-one correspondence between rational multiplicative group actions on an algebraic variety $X$ and derivations $\partial\colon K_X\to K_X$ of the field of fractions $K_X$ of $X$ satisfying that there exists a generating set $\{a_i\}_{i\in I}$ of $K_X$ as a field such that $\partial(a_i)=λ_i a_i$ with $λ_i \in \mathbb{Z}$ for all $i\in I$. We call such derivations rational semisimple. Furthermore, we also prove the existence of a rational slice for every rational semisimple derivation, i.e., an element $s\in K_X$ such that $\partial(s)=s$. By analogy with the case of additive group actions case, we prove that $K_X\simeq K_X^{\mathbb{G}_m}(s)$ and that under this isomorphism the derivation $\partial$ is given by $\partial=s\frac{d}{ds}$. Here, $K_X^{\mathbb{G}_m}$ is the field of invariant of the $\mathbb{G}_m$-action.

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arXiv2022

Cohen-Macaulay Du Bois singularities with a torus action of complexity one

Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via divisorial fans and Kóvacs-Schwede-Smith characterization of Du Bois singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an example of a Cohen-Macaulay Du Bois singularity of complexity one which does not have rational singularities.

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arXiv2022

Characterization of affine surfaces with a torus action by their automorphism groups

In this paper we prove that if two normal affine surfaces $S$ and $S'$ have isomorphic automorphism groups, then every connected algebraic group acting regularly and faithfully on $S$ acts also regularly and faithfully on $S'$. Moreover, if $S$ is non-toric, we show that the dynamical type of a 1-torus action is preserved in presence of an additive group action. We also show that complex affine toric surfaces are determined by the abstract group structure of their regular automorphism groups in the category of complex normal affine surfaces using properties of the Cremona group. As a generalization to arbitrary dimensions, we show that complex affine toric varieties, with the exception of the algebraic torus, are uniquely determined in the category of complex affine normal varieties by their automorphism groups seen as ind-groups.

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arXiv2021

On the liftability of the automorphism group of smooth hypersurfaces of the projective space

Let $X$ be a smooth hypersurface of dimension $n\geq 1$ and degree $d\geq 3$ in the projective space given as the zero set of a homogeneous form $F$. If $(n,d)\neq (1,3), (2,4)$ it is well known that every automorphism of $X$ extends to an automorphism of the projective space, i.e., $\operatorname{Aut}(X)\subseteq \operatorname{PGL}(n+2,\mathbb{C})$. We say that the automorphism group $\operatorname{Aut}(X)$ is $F$-liftable if there exists a subgroup of $\operatorname{GL}(n+2,\mathbb{C})$ projecting isomorphically onto $\operatorname{Aut}(X)$ and leaving $F$ invariant. Our main result in this paper shows that the automorphism group of every smooth hypersurface of dimension $n$ and degree $d$ is $F$-liftable if and only if $d$ and $n+2$ are relatively prime. We also provide an effective criterion to compute all the integers which are a power of a prime number and that appear as the order of an automorphism of a smooth hypersurface of dimension $n$ and degree $d$. As an application, we give a sufficient condition under which some Sylow $p$-subgroups of $\operatorname{Aut}(X)$ are trivial or cyclic of order $p$.

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