Blanc, Jérémy; Déserti, Julie: Embeddings of SL(2,Z) into the Cremona group, , 2012

Geometric and dynamic properties of embeddings of SL(2;Z) into the Cremona group are studied. Infinitely many

non-conjugate embeddings which preserve the type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements of

the same type) are provided. The existence of infinitely many non-conjugate elliptic, parabolic and hyperbolic embeddings is also

shown.

In particular, a group G of automorphisms of a smooth surface S obtained by blowing-up 10 points of the complex projective

plane is given. The group G is isomorphic to SL(2;Z), preserves an elliptic curve and all its elements of infinite order are hyperbolic.

Gorla, Elisa; Migliore, Juan; Nagel, Uwe: Groebner bases via linkage, , 2012

In this paper, we give a sufficient condition for a set G of polynomials to be a Groebner basis with respect to a given term-order for the ideal I that it generates. Our criterion depends on the linkage pattern of the ideal I and of the ideal generated by the initial terms of the elements of G. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder, where the size of the minors or pfaffians is allowed to vary in different regions of the matrix or the ladder. We use the sufficient condition that we established to prove that the minors or pfaffians form a reduced Groebner basis for the ideal that they generate, with respect to any diagonal or anti-diagonal term-order. We also show that the corresponding initial ideal is Cohen-Macaulay. Our proof relies on known results in liaison theory, combined with a simple Hilbert function computation. In particular, our arguments are completely algebraic.

Blanc, Jérémy; Lamy, Stéphane: Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links, , 2012

We characterise smooth curves in P3 whose blow-up produces

a threefold with anticanonical divisor big and nef. These are

curves C of degree d and genus g lying on a smooth quartic, such that

(i) 4d 􀀀 30  g  14 or (g; d) = (19; 12), (ii) there is no 5-secant line,

9-secant conic, nor 13-secant twisted cubic to C. This generalises the

classical similar situation for the blow-up of points in P2.

We describe then Sarkisov links constructed from these blow-ups, and

are able to prove the existence of Sarkisov links which were previously

only known as numerical possibilities.

Kraft, Hanspeter; Stampfli, Immanuel: Automorphisms of the affine Cremona group, , 2011

Blanc, Jérémy: Elements and cyclic subgroups of finite order of the Cremona group, , 2011

We give the classication of elements { respectively cyclic subgroups {

of nite order of the Cremona group, up to conjugation. Natural parametri-

sations of conjugacy classes, related to xed curves of positive genus, are

provided.

De Negri, Emanuela; Gorla, Elisa: Invariants of ideals generated by pfaffians, , 2011

Ideals generated by pfaffians are of interest in commutative algebra and algebraic geometry, as well as in combinatorics. In this article we compute multiplicity and Castelnuovo-Mumford regularity of pfaffian ideals of ladders. We give explicit formulas for some families of ideals, and indicate a procedure that allows to recursively compute the invariants of any pfaffian ideal of ladder. Our approach makes an essential use of liaison theory.

Dubouloz, Adrien; Moser-Jauslin, Lucy; Poloni, Pierre-Marie; : Noncancellation for smooth contractible affine threefolds, , 2011

Blanc, Jérémy: The best polynomial bounds for the number of triangles in a simple arrangement of n pseudo-lines, , 2011

It is well-known and easy to observe that affine (respectively projective)

simple arrangement of n pseudo-lines may have at most n(n − 2)/3

(respectively n(n − 1)/3) triangles. However, these bounds are reached for

only some values of n (mod 6). We provide the best polynomial bound for the

affine and the projective case, and for each value of n (mod 6).

Kraft, Hanspeter; Kohls, Martin: Degree bounds for separating invariants, , 2010

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v').
It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree ≤ |G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist.
If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) ≤ b(G) ≤ [G:H] b(H), and that b(G) ≤ b(G/H) b(H) in case H is normal. Moreover, we calculate b(G) for some specific finite groups.

Dubouloz, Adrien; Moser-Jauslin, Lucy; Poloni, Pierre-Marie; : Inequivalent embeddings of the Koras-Russell cubic threefold, , 2010

Kraft, Hanspeter; Lötscher, Roland; Schwarz, Gerald W.: Compression of finite group actions and covariant dimension, II, , 2009

This is a continuation of a previous paper with the same title. In that paper we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of that paper, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps.

Blanc, Jérémy: Linearisation of finite Abelian subgroups of the Cremona group of the plane, , 2009

Given a finite Abelian subgroup of the Cremona group of the plane, we

provide a way to decide whether it is birationally conjugate to a group of

automorphisms of a minimal surface.

In particular, we prove that a finite cyclic group of birational transformations

of the plane is linearisable if and only if none of its non-trivial elements

fix a curve of positive genus. For finite Abelian groups, there exists only one

surprising exception, a group isomorphic to Z/2Z xZ/4Z, whose non-trivial

elements do not fix a curve of positive genus but which is not conjugate to

a group of automorphisms of a minimal rational surface.

We also give some descriptions of automorphisms (not necessarily of finite

order) of del Pezzo surfaces and conic bundles.

Blanc, Jérémy; Pan, Ivan; Vust, Thierry: On birational transformations of pairs in the complex plane, , 2009

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.

Bonnet, Philippe; Vénéreau, Stéphane: Relations between the leading terms of a polynomial automorphism, , 2009

Let I be the ideal of relations between the leading terms of the polynomials defining an automorphism of Kⁿ. In this paper, we prove the existence of a locally nilpotent derivation which preserves I. Moreover, if I is principal, i.e. I=(R), we compute an upper bound for deg₂(R) for some degree function deg₂ defined by the automorphism. As applications, we determine all the principal ideals of relations for automorphisms of K³ and deduce two elementary proofs of the Jung-van der Kulk Theorem about the tameness of automorphisms of K².

Blanc, Jérémy: The correspondence between a plane curve and its complement, , 2009

Given two irreducible curves of the plane which have isomorphic complements, it is natural to ask whether there exists an automorphism of the plane that sends one curve on the other.

This question has a positive answer for a large family of curves and H. Yoshi- hara conjectured that it is true in general. We exhibit counterexamples to this conjecture, over any ground field. In some of the cases, the curves are isomor- phic and in others not; this provides counterexamples of two different kinds.

Finally, we use our construction to find the existence of surprising non- linear automorphisms of affine surfaces.

Benedetti, Bruno; Constantinescu, Alexandru; Varbaro, Matteo: Dimension, Depth and Zero-Divisors of the Algebra of Basic k-Covers of a Graph, , 2008

We study the basic k-covers of a bipartite graph G; the algebra ¯A(G) they span, first studied
by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when
¯A(G) is a domain in terms of the combinatorics of G; if follows from a result of Hochster
that when ¯A(G) is a domain, it is also Cohen-Macaulay. We then study the dimension of
¯A(G) by introducing a geometric invariant of bipartite graphs, the “graphical dimension”.
We show that the graphical dimension of G is smaller than or equal to dim(¯A(G)), and
equality holds in many cases (e.g. when G is a tree, or a cycle). Finally, we discuss
applications of this theory to the arithmetical rank.

Bartholdi, Nicolas; Blanc, Jérémy; Loisel, Sébastien: On simple arrangements of lines and pseudo-lines in P2 and R2 with the maximal number of triangles, , 2008

Blanc, Jérémy; Pan, Ivan; Vust, Thierry: Sur un théorème de Castelnuovo, , 2008

We continue the study of G. Castelnuovo on the group of birational

transformation of the complex plane that fix each point of a curve of genus

> 1 ; we use adjoint linear system of the curve as Castelnuovo does.

We prove that these groups are abelian, and that these are either finite, of

order 2 or 3, or conjuguate to a subgroup of the de Jonqui`eres group. We show

also that these results do not generalise to curves of genus  1.

Keywords. Cremona transformations, birational transformations, fixed curves,

curves of big genus, adjoint linear system, de Jonqui`eres transformations.

Favi, Giordano; Florence, Mathieu: Tori and essential dimension, , 2008

Balmer, Paul; Favi, Giordano: Gluing techniques in triangular geometry, , 2007

van den Essen, Arno; Maubach, Stefan; Vénéreau, Stéphane: The special automorphism group of R[[t](t(m))[x(1),..., x(n)] and coordinates of a subring of R[t][x(1), . . ., x(n)], , 2007

Let R be a ring. The Special Automorphism Group SAut(R)R[x(1),..., x(n)] is the set of all automorphisms with determinant of the Jacobian equal to 1. It is shown that the canonical map of SAut(R[t]) R[t][x(1),...,x(n)] to SAutR(m) R-m[x(1),...,x(n)] where R-m := R[t]/(t(m)) and Q subset of R is surjective. This result is used to study a particular case of the following question: if A is a subring of a ring B and f is an element of A vertical bar n vertical bar is a coordinate over B does it imply that f is a coordinate over A? It is shown that if A = R[t(m), t(m+1),...] subset of R[t] = B then the answer to this question is ``yes``. Also, a question on the Venereau polynomial is settled, which indicates another ``coordinate-like property`` of this polynomial. (c) 2006 Elsevier B.V. All rights reserved.

Blanc, Jérémy: Conjugacy classes of affine automorphisms of Kn and linear automorphisms of Pn in the Cremona groups, , 2006

We describe the conjugacy classes of affine automorphisms in the group

Aut(n,K) (respectively Bir(Kn)) of automorphisms (respectively of birational

maps) of Kn. From this we deduce also the classification of conjugacy classes of

automorphisms of Pn in the Cremona group Bir(Kn).

Vénéreau, Stéphane: New bad lines in R[x, y] and optimization of the Epimorphism Theorem, , 2006

We give necessary and sufficient conditions on a commutative ring with unity R so that the following holds: any polynomial f in R[x, y] such that R[x, y]/(f) similar or equal to R-[1] (a line for short) is a variable (= alpha(x) for some automorphism alpha:R[x, y] -> R[x, y]). The main ingredient here is the construction of new bad lines (lines which are not variables). (c) 2006 Elsevier Inc. All rights reserved.

Kraft, Hanspeter; Wallach, Nolan: On the nullcone of representations of reductive groups, , 2006

We study the geometry of the nullcone N(V^k) for several copies of a representation V of a reductive group G and its behavior for different k. We show that for large k there is a certain "stability" with respect to the irreducible components. In the case of the so-called theta-representations, this can be made more precise by using the combinatorics of the weight system as a subset of the root system. All this finally allows to calculate explicitly and in detail a number of important examples, e.g. the cases of 3- and 4-qubits which play a fundamental role in quantum computing.

Vénéreau, Stéphane: Hyperplanes of the form f(1) (x, y)z(1)+center dot center dot center dot+f(k)(x, y)z(k)+g(x, y) are variables, , 2005

The Abhyankar-Sathaye Embedded Hyperplane Problem asks whether any hypersurface of U isomorphic to Cn-1 is rectifiable, i.e., equivalent to a linear hyperplane up to an automorphism of C-n. Generalizing the approach adopted by Kaliman, Venereau, and Zaidenberg, which consists in using almost nothing but the acyclicity of Cn-1, we solve this problem for hypersurfaces given by polynomials of C[x, y, z(1),..., z(k)] as in the title.

Beauville, Arnaud; Blanc, Jérémy: On Cremona transformations of prime order, , 2004

We prove that an automorphism of order 5 of the Del Pezzo surface S5 of degree 5 is conjugate through a

birational map S 99K P2 to a linear automorphism of P2. This completes the classification of conjugacy classes of

elements of prime order in the Cremona group.

Berhuy, Grégory; Favi, Giordano: Essential dimension: a functorial point of view (after A. Merkurjev), , 2003

Kaliman, Shulim; Vénéreau, Stéphane; Zaidenberg, Mihail: Extensions birationnelles simples de l'anneau de polynômes $C^{[3]}$, , 2001

The Abhyankar-Sathaye Problem asks wether any biregular embedding
of C^k into Cⁿ can be rectified. Here we study
this problem for the embeddings of C³ into C⁴ whose image X is
given in C⁴ by an equation p = f(x; y)u+g(x; y; z) = 0. Under certain additional assumptions we show that, indeed,
the polynomial p is a variable of the polynomial ring C^[4] = C[x; y; z; u] (i.e., a
coordinate of a polynomial automorphism of C⁴). This is an analog of a theorem
due to Sathaye (1976) which concerns the case of embeddings of C² into C³.
Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial
p as above, a criterion for when X is isomorphic to C³.

Kraft, Hanspeter; Small, Lance W.; Wallach, Nolan R.: Properties and examples of FCR-algebras, , 2001

An algebra A over a field k is FCR if every finite dimensional representation of A is completely reducible and the intersection of the kernels of these representations is zero. We give a useful characterization of FCR-algebras and apply this to C*-algebras and to localizations. Moreover, we show that ``small'' products and sums of FCR-algebras are again FCR.

Kraft, Hanspeter; Schwarz, Gerald W.: Representations with a Reduced Null Cone, 2011

Favi, Giordano: Quadratic Forms and Linear Algebraic Groups, 2009

Topics discussed at the workshop Quadratic Forms and Linear Algebraic Groups included besides the algebraic theory of quadratic and Hermitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic aspects pertaining to the theory of quadratic forms over function fields over local fields.