Xavier Roulleau

Negative curves on algebraic surfaces

We study curves of negative self-intersection on algebraic surfaces. Our main result shows there exist smooth complex projective surfaces X, related to Hilbert modular surfaces, such that X contains reduced, irreducible curves C of arbitrarily negative self-intersection C 2 . Previously the only known examples of surfaces for which C 2 was not bounded below were in positive characteristic, and the general expectation was that no examples could arise over the complex numbers. Indeed, we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field, and thus our complex examples require a different approach.

Chern slopes of simply connected complex surfaces of general type are dense in (2,3)

We prove that for any number

Elliptic curve configurations on Fano surfaces

r

On Schoen surfaces

in

L’application cotangente des surfaces de type général

[2,3]

Lines on Cubic Hypersurfaces Over Finite Fields

, there are spin (resp. non-spin minimal) simply connected complex surfaces of general type

On the hyperbolicity of surfaces of general type with small c12

X

The Fano surface of the Fermat cubic threefold, the del Pezzo surface of degree

with

5

c_1^2(X)/c_2(X)

and a ball quotient

arbitrarily close to