Dino Lorenzini

An invitation to arithmetic geometry

Integral closure Plane curves Factorization of ideals The discriminants The ideal class group Projective curves Nonsingular complete curves Zeta-functions The Riemann-Roch Theorem Frobenius morphisms and the Riemann hypothesis Further topics Appendix Glossary of notation Index Bibliography.

A finite group attached to the laplacian of a graph

Abstract Let F = diag(ϕ1, ⋯ ϕn−1, 0), ϕ1 | ⋯ | ϕn-1, denote the Smith normal form of the laplacian matrix associated to a connected graph G on n vertices. Let h denote the cardinal of the set {i | ϕi > 1}. We show that h is bounded by the number of independent cycles of G and we study some cases where these two integers are equal.

Smith normal form and Laplacians

Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group @F(G), obtained from the Smith normal form of M, and whose order is the number of spanning trees of G. We provide some general results on the relationship between the eigenvalues of M and the structure of @F(G), and address the question of how often the group @F(G) is cyclic.

Models of Curves and Finite Covers

Let K be a discrete valuation field with ring of integers O K .Letf : X ! Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over O K of X and of Y. Three such relationships are listed below. Consider a Galois cover f : X ! Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K,thenX achieves semi-stable reduction over some explicit tame extension of K.B/.WhenK is strictly henselian, we determine the minimal extension L=K with the property that X L has semi-stable reduction. Let f : X ! Y be a finite morphism, with g.Y/ > 2. We show that if X has a stable model X over O K ,thenY has a stable model Y over O K , and the morphism f extends to a morphism X ! Y. ! Y. Finally, given any finite morphism f : X ! Y, is it possible to choose suitable regular models X and Y of X and Y over O K such that f extends to a finite morphism X ! Y ?As wasshown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situ-ations, with f a cyclic cover of any order > 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.

Arithmetical properties of laplacians of graphs

Let denote any matrix. Thinking of M as a linear map , we denote by Im(M) the -span of the column vectors of M. Let e 1, …en denote the standard basis of and let Eij :=ei − ej , (i ν j). In this article, we are interested in the group , and in particular in the elements of this group defined by the images τ ij of the vectors Eij under the quotient . Most of this article is devoted to the study of the case where M is the laplacian of a graph. In this case, the elements τ ij have finite order, and we study how the geometry of the graph relates to these orders. We give in particular a criterion in terms of the topology of the graph to determine when such an element has order 1 or 2.

Hypersurfaces in projective schemes and a moving lemma

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subscheme C, and intersecting properly a closed set F. Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied if R is the ring of integers of a number field, or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1-cycles on a regular scheme X quasi-projective and flat over S. We also show the existence of a finite surjective S-morphism to the projective space P_S^d for any scheme X projective over S when X/S has all its fibers of a fixed dimension d.

Corrigendum to Néron models, Lie algebras, and reduction of curves of genus one and The Brauer group of a surface

Let X be a proper smooth and connected surface over a finite field. We proved in [LLR2] that the order of the Brauer group Br(X) of X is a perfect square if it is finite. Our proof is based in part on a result of Gordon [Gor], which we used in [LLR1] to establish a key formula. Thomas Geisser noted that the formula in [LLR1] is incorrect, due to an omission in [Gor], and provides a corrected formula. We explain in this corrigendum how to modify the work of Gordon to establish a correct formula. The corrected formula can be used to prove the result in [LLR2] without further modifications.