Ignacio Sols

Moduli spaces for principal bundles in arbitrary characteristic

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic

Constraining the Kähler Moduli in the Heterotic Standard Model

Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitly, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable.

The Severi bound on sections of rank two semistable bundles on a Riemann surface

Let C be a compact Riemann surface of genus g≥1 . For a line bundle L on C of degree d with 0≤d≤2g−2 , the classical theorem of Clifford states that the dimension h 0 (L) of the space of global sections of L satisfies h 0 (L)≤d 2 +1 . It is also quite easy to list the cases which give equality (apart from the trivial bundle and the canonical bundle, these exist only on hyperelliptic curves). This theorem can be extended to a semistable bundle E on C of rank n and degree d with 0≤d≤n(2g−2) , when it states that h 0 (E)≤d 2 +n . Now let E be a semistable bundle of rank 2 . In this case the authors obtain a more refined result depending on the Segre invariant s(E) , which is defined as the minimum value of degE/L−degL for L a line subbundle of E . (In particular, the semistability condition becomes s(E)≥0 .) The main theorem is that, if degE=d and s≤d≤4g−4−s , then h 0 (E)≤d−s 2 +δ , where δ=1 or δ=2 . (Note that d−s is always even.) In a previous paper, E. Arrondo and the second author conjectured this inequality with δ=1 (with a small number of exceptions similar to those for rank 1 ). However, this is not correct and δ=2 is sometimes necessary; the authors call the bundles which attain this bound Severi bundles. The number δ is obtained as follows. We define polynomials K r (n,N) (the Krawtchouk polynomials) by means of the generating function ∑ r K r (n,N)z r =(1−z) n (1+z) N−n . Let r=d−s 2 +1 ; then we can take δ=1 if K r (g,2g−s)≠0 and δ=2 otherwise. The connection with Krawtchouk polynomials is intriguing, since these are used in coding theory and recent work of T. Johnsen has established links between coding theory and semistable bundles of rank 2 on Riemann surfaces; these links make essential use of the Segre invariant s . The key to the proof of the inequality is the following lemma. Let E be a bundle of rank 2 and degree d on a curve of genus g having at least r+1≤g independent sections. If K r (g,2g+2r−2−d)≠0, then there is a nonzero section of E vanishing at r points. This is proved using a computation of Chern classes, formal properties of K r (n,N) and the Porteous formula. The authors give examples to show that, for any allowable values of g,s,d , the bound δ=1 can be attained (in fact on a hyperelliptic curve). A good deal is known about the vanishing of K r (n,N) , thus providing a restricted list of candidates for the invariants of Severi bundles, but these values do not all arise from actual bundles. The Severi bundles with d−s=0 are all known, and the authors compute those with d−s=2 and d−s=4 . In particular the bundles obtained by I. Grzegorczyk [Ulam Quart. 3 (1996), no. 2, 41 ff., approx. 3 pp. (electronic) ] are Severi bundles with g=5 , s=4 , d=8 . The authors also obtain sharp bounds for the numbers of independent sections of non-semistable bundles.

STABILITY OF CONIC BUNDLES: WITH AN APPENDIX BY MUNDET I RIERA

Roughly speaking, a conic bundle is a surface, fibered over a curve, such that the fibers are conics (not necessarily smooth). We define stability for conic bundles and construct a moduli space. We prove that (after fixing some invariants) these moduli spaces are irreducible (under some conditions). Conic bundles can be thought of as generalizations of orthogonal bundles on curves. We show that in this particular case our definition of stability agrees with the definition of stability for orthogonal bundles. Finally, in an appendix by I. Mundet i Riera, a Hitchin-Kobayashi correspondence is stated for conic bundles.

A GIT interpretation of the Harder–Narasimhan filtration

An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a “maximally destabilizing” 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder–Narasimhan filtration.

Bounding families of ruled surfaces

In this paper we provide a sharp bound for the dimension of a family of ruled surfaces of degree d in P3 K. We also _nd the families with maximal dimension: the family of ruled surfaces containing two unisecant skew lines, when d _ 9 and the family of rational ruled surfaces, when d _ 9. The first tool we use is a Castelnuovo-type bound for the irregularity of ruled surfaces in Pn K. The second tool is an exact sequence involving the normal sheaf of a curve in the grassmannian. This sequence is analogous to the one constructed by Eisenbud and Harris in 1992, where they deal with the problem of bounding families of curves in projective space. However, our construction is more general since we obtain the mentioned sequence by purely algebraic means, studying the geometry of ruled surfaces and of the grassmannian.

Connectedness of intersections of special Schubert varieties

Let Gr l,n be the Grassmann variety of l -dimensional subspaces of an n -dimensional vector space V over an algebraically closed field k . Let σ(W)={Λ∈Gr l,n : Λ∩W≠0} denote the special Schubert variety associated to a subspace W of V . The main theorem of the paper is the following: The intersection ⋂ m j=1 σ(V j ) of the special Schubert varieties associated to subspaces V j , j=1,2,⋯,m , of dimension n−l−a j +1 such that l(n−l)−∑ m j=1 a j >0 is connected. Moreover, the intersection is irreducible of dimension l(n−l)−∑ m j=1 a j for a general choice of V j . The authors conjecture that the irreducibility holds for intersections of arbitrary Schubert varieties, when they are in general position with nonempty intersection. For a related connectivity result the authors refer to a paper of J. P. Hansen [Amer. J. Math. 105 (1983), no. 3, 633–639].

Stable Higgs G-sheaves

For a connected reductive group G, we generalize the notion of (semi)stable Higgs G-bundles on curves to smooth projective schemes of higher dimension, allowing also Higgs G-sheaves, and construct the corresponding moduli space.

Stable vector bundles and string theory

In [4], Braun, He, Ovrut and Pantev proposed a model of string theory (based on the Calabi‐Yau 3‐fold X defined in [3]) whose low energy limit predicts certain properties of the Standard Model of particle Physics. This model depends on two vector bundles that have to be stable. We calculate the ample cone of X, and prove that one of them is stable, and the other one is not.

The Schubert Triangle Geometry on an Algebraic Surface

For a smooth complex projective surface, and for two families of curves with traditional singularities in it, we enumerate the pairs of curves in each family having two points of contact among them, thus generalizing the double contact formulae known or conjectured by Zeuthen and Schubert in the case of the complex projective plane. The technique we use to this purpose is a particular notion of triangle which can be defined in any smooth surface, thus potentially generalizing to arbitrary surfaces the Schubert technique of triangles.