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Nicolas Perrin

Publications and source records attributed to Nicolas Perrin.

65 records · Page 4Linked to original sources
arXiv2006

Small resolutions of minuscule Schubert varieties

In this paper, we describe on the one hand, all relative minimal models Y of a minuscule Schubert variety X using some combinatorics on quivers and we prove that the morphism from Y to X is small (in the sense of intersection cohomology). On the other hand, thanks to a result of B. Totaro, any small resolution Z of a minuscule Schubert variety X has to be a relative minimal model of X. So X admits a small resolution if and only if there exists a smooth relative minimal model Y of X. We give a combinatoric criterion for Y to be smooth describing in this way all small resolutions of X. We also use stringy polynomials and the relative canonical model to give another way to tell when a minuscule Schubert variety admits a small resolution.

math.AG↗
arXiv2004

Rational curves on minuscule Schubert varieties

Let X be a minuscule Schubert variety and $α$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $α$ from a rational curve to X. The irreducible components are described in the following way : the class $α$ can be seen as an element of $Pic(X)^*$ the dual of the Picard group. Because any Weil-divisor need not to be a Cartier-divisor, there is (only) a surjective map $s:A^1(X)^*\to Pic(X)^*$ from the dual of the group of codimension 1 cycles to the dual of the Picard group. The irreducible components are given by the effective elements $β$ in $A^1(X)^*$ such that $s(β)=α$. The proof of the result uses the Bott-Samelson resolution Y of X. We prove that any curve on X can be lifted in Y (after deformation). This is because any divisor on minuscule Schubert variety is a moving one. Then we prove that any curve coming from X can be deformed so that it does not meet the contracted divisor of $Y\to X$. This is possible because for minuscule Schubert variety there are lines in the projectivised tangent space to a singularity. It is now sufficient to deal with the case of the orbit of $Stab(X)$ the stabiliser of X and we can apply results of our previous paper math.AG/0003199.

math.AG↗
arXiv2004

Rational curves on homogeneous cones

Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety. Let $α$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $α$ from a rational curve to X. The situation depends on the line bundle L : if the projectivised tangent space to the vertex contains lines (i.e. if G/Q contains lines in P) then the irreducible components are described as in our paper math.AG/0407123 by the difference between Cartier and Weil divisors. On the contrary if there is no line in the projectivised tangent space to the vertex then there are new irreducible components corresponding to the multiplicity of the curve through the vertex. As in math.AG/0407123 we use a resolution Y of X (the blowing-up) and study the curves on Y.

math.AG↗
arXiv2003

Quelques remarques sur les courbes de genre 5

In this article we study genus 5 curves with a fixed point free involution. We give two geometrical caracterisations of these curves amoung all genus 5 curves. One of these was conjectured by Arbarello, Cornalba, Griffiths and Harris in their book on curves. The second one is given by some particular embeddings corresponding to the points of the Prym variety associated to the situation.

math.AG↗
arXiv2001

Deformation de fibres vectoriels sur les varietes lisses de dimension trois

Let X be a smooth 3-fold and let E be a rank 2 torsion free sheaf. In the first part of this paper, we give some necessary conditions for the sheaf E to be limit of vector bundles. In the second part, we describe an example of this problem. Thanks to the construction of Ellingsrud and Stromme, we describe a familly of rank 2 sheaves on P^3 (the 3-dimensional projective space). We show that for this familly, the conditions of the first part are sufficient.

math.AG↗
arXiv2001

Limites de fibres vectoriels dans ${\bf M}_{\mathbb{Q}_3}(0,2,0)$

Let $\mathbb{Q_3}$ the smooth quadric in P^4 (the 4-dimensional projective space). In this note we completely describe the closure of the open set of vector bundles in ${\bf M}_{\mathbb{Q}_3}(0,2,0)$. This description gives another example where the conditions given in math.AG/0112199 for a sheaf to be limit of vector bundles are necessary and sufficient.

math.AG↗
arXiv2001

Two components of the boundary of the compactification of the variety of instantons

We study two components of the boundary of the compactification of the variety I_3 of instantons of degree three. We use the desciption of I_3 as symetric (involutive) cubo-cubic transforms deduced from the Beilinson monade. It involves some geometry of curves and surfaces in P^3. This allows us to distinguish two irreducible components which are in the closure of involutive cubo-cubic transforms. It gives us two irreducible components of the boundary of I_3. Moreover, we show that the cubo-cubic transforms of one of these components are the inverse of the other one.

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arXiv2001

Singular locus of rational ruled surfaces

We prove that the morphism that maps a rational ruled surface to its singular locus is genericaly injective modulo isomophism and duality. We also calculate the dimension and the degre of its image.

math.AG↗
arXiv2000

Courbes rationnelles sur les variétés homogènes et une désingularisation plus fine des variétés de Schubert

In this article we prove the irreducibility of the Hilbert scheme of rationnal curves on homogeneous varieties with fixed class in the Chow ring. This result has also been proved by J. F. Thomsen [T] and B. Kim and R. Pandharipande [KP]. Our method is totaly different (we don't use the compactification of stable maps) and enables us to prove the existence of rational smooth curves on homogeneous varities with fixed class in the Chow ring. This was not the case of Thomsen's and Kim and Pandharipande's proofs. We use a decomposition of G/P in orbits (called the P'-orbits, see definition) which are bigger than the Schubert cells. We then prove that these P'-orbits are "towers" of affine bundles (see definition) over "smaller" homogeneous varities. This description gives the results. Our decomposition in P'-orbits enables us to give a "better" desingularisation of Schubert varities than Demazure's one.

math.AG↗
arXiv2000

Eclatement de réseaux de quadriques et bord des instantons de degré 3

In their article [1], L. Gruson and M. Skiti have constructed a birationnal map from the variety $\I$ of mathematical instantons of degree 3 to the variety of nets of quadrics in $\pd$. They describe by this way two irreducible componants of the boundary of $\I$ associated to the divisor of nets which contain a two-plane degenerated quadric and the divisor of L\" uroth nets. In this article we describe an irreducible componante of the boundary of $\I$ as the exceptionnal divisor of the blowing-up of the closed set of nets of quadrics of rank 3.

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