On degeneracy loci of equivariant bi-vector fields on a smooth toric variety

We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension $n$, the locus where the rank of an equivariant bi-vector field is $\leq 2k$ is not empty and has at least a component of dimension $\geq 2k+1$, for all integers $k> 0$ such that $2k < n$. The same is true also for $k=0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.Comment: 11 page

A short note on infinity-groupoids and the period map for projective manifolds

A common criticism of infinity-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naive intuition of what an infinity-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths' expression for the differential of the periods map, the Kodaira principle on obstructions to deformations of projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson quasi-abelianity theorem are easily recovered.Comment: 13 pages; uses xy-pic; exposition improved and a few inaccuracies corrected; an hypertextual version of this article is available at http://ncatlab.org/publications/published/FiorenzaMartinengo201

Mori Dream Stacks

We propose a generalisation of Mori dream spaces to stacks. We show that this notion is preserved under root constructions and taking abelian gerbes. Unlike the case of Mori dream spaces, such a stack is not always given as a quotient of the spectrum of its Cox ring by the Picard group. We give a criterion when this is true in terms of Mori dream spaces and root constructions. Finally, we compare this notion with the one of smooth toric Deligne-Mumford stacks.Comment: 19 pages, to appear in Mathematische Zeitschrif

Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees with a noncommutative Cech cohomology-type functor.Comment: Several typos corrected. 22 pages, uses xy-pi

It Has Always Called Me: Answering the Call

Con las amigas Laura Minguzzi, Marina Santini y Luciana Tavernini, al final de los años ochenta fundamos en la Librería delle donne di Milano una Comunidad, la Comunidad de práctica y reflexión pedagógica y de investigación histórica, en la cual pensábamos yconstruíamos políticamente en un primer tiempo la pedagogía y en un segundo tiempo la historia. En 2001, en Milán, organizamos también un congreso nacional: Cambia el mundo, cambia la historia. La diferencia sexual en la investigación y en la enseñanza. La historiadora María-Milagros Rivera Garretas, en el XII “Simposiode la Asociación Internacional de Filósofas” mostró la importancia de dar un nuevo inicio a la historia sacando a la luz los nudos no resueltos de la propia historia; porque indagándolos, soltando lo no dicho, es posible encontrar las palabras que explican, coherentemente contigo, la experiencia femenina. En otoño de 2006, tranquilizada ya sobre el sentido y el significado de mi libro sobre la abuela, propuse que cambiáramos el nombre de nuestra Comunidad y la llamásemos Comunidad de Historia Viviente. Ladenominación tenía su origen en las palabras de mi La voce del silenzio: “Hay una historia viviente anidada en cada una y cada uno de nosotras...”. El cambio de nombre significaba esencialmente la introducción de un modo nuevo de “hacer historia”: tomando prestado el proceso que me había llevado a extraer de mí la historia de la abuela sustraída, cada una buscaría dentro de sí un nudo no resuelto, el enredo oscuro no investigado que hay en el interior de cada cual: habríamos empezado a escribir lo que se siente. O sea,historia verdadera, según la enseñanza de María Zambrano. Escarmentada de mi sufrimiento, mi intención era también la de hacer que ellas recorrieran en intercambio relacional ese camino que para mí había sido solitario y melancólico. Las amigas aceptaron y empezó así una práctica no experimentada antes: la práctica de la historia viviente.With friends Laura Minguzzi, MarinaSantini and Luciana Tavernini, at the end of the eightieswe founded a Community at the Milan Women’sBookstore, the Community of pedagogical practice andreflection, in which we thought and built politically,first teaching and learning, and then history. In 2001, inMilan, we also organised a national conference: The WorldChanges, History Changes. Sexual Difference in Research andTeaching. The historian María-Milagros Rivera Garretas,in the XII “Symposiom of the International Associationof Women Philosophers” showed the importance ofgiving a new beginning to history, bringing into the lightthe unresolved knots of one’s own history; because inresearching into them, releasing that which was previouslyunsaid, it is possible to find the words that explain, incoherence with yourself, female experience. In the autumnof 2006, by then at peace with the sense and meaning ofmy book on my grandmother, I proposed that we changethe name of our Community and that we call it theCommunity of Living History. Its denomination has itsroots in the words of my La voce del silenzio: “There is a living history nesting in each one of us...”. The changeof name meant in essence the intrduction of a new wayof “doing history”: borrowing from the process that hadled me to extract from myself the history of my abductedgrandmother; each one would look inside herself for anunresolved knot, the dark unresearched tangle that thereis inside everyone: we would have begun to write whatone feels. That is, true history, according to the teachingsof María Zambrano. Wary of my suffering, my intentionwas also that of making them walk that path, which forme had been solitary and sad, in a relational exchange. Myfriends accepted and thus began a practice previously notexperienced: the practice of living history

Local structure of Brill-Noether strata in the moduli space of flat stable bundles

We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kahler manifold, proving that these strata have quadratic algebraic singularities

From the Linguistic Turn to the Pictorial Turn - Hermeneutics Facing the "Third Copernican Revolution"

n this article, I will deal with the role of pictures within discus- sions on the crisis of contemporary philosophical hermeneutics. Firstly, I will argue that some of the aporias this debate runs up against can be over- come by a reconsideration of the traditional dualism of word and image. In this regard my reference point is Paul Ricoeur’s theory of metaphor. His model of the living metaphor overcomes the difficulties posed by the hermeneutics of the symbol and restores the image to an internal force within language: image seeks language to give itself form, and language seeks image to feed on new meanings. Secondly, I will reconstruct some aspects of the interdisciplinary debate around the so-called ‘pictorial turn’, in which, I will argue, very interesting features can be identified that Ri- coeur’s theory of metaphor merely foreshadows. In this reactivation of the role of pictures, the debate concerning the outcomes of twentieth-century hermeneutics can probably make new claims on our attention

Los paisajes italianos de Quevedo

Lejos de compartir la opinión según la cual en la poesía y las prosas de Quevedo el paisaje (en este caso el italiano) no habría dejado la menor huella, consideramos que, al menos bajo ciertas condiciones, el escritor participa emotivamente en la visión de lo real, aunque siempre partiendo de un acto o estímulo intelectual, conceptual, que le lleva ora a descubrir una paradoja o antinomia en lo que ve (así por ejemplo en su descripción de Génova), ora a entreverar en su visión agudezas o reminiscencias eruditas (así por ejemplo en la descripción de Roma o de los volcanes del sur de Italia)