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 Titles and Abstracts Carolina Araujo - Explicit log Fano structures on blowups of projective spaces In this talk we will discuss blowups of projective spaces at general points. We will determine which of these blowups $X$ are log Fano, i.e., when there is a $\mathbb{Q}$-divisor $\Delta$ on $X$ such that $(X, \Delta)$ is klt and $-(K_X+\Delta)$ is ample. This is a joint work with Alex Massarenti. Gilberto Bini - A New Calabi-Yau Threefold In this talk we will describe the geometry of a new Calabi-Yau threefold Y that is obtained as a crepant resolution of a complete intersection with 72 ordinary double points in five dimensional projective space. The Picard number and the number of moduli of Y coincide and they are both equal to 10. This is joint work with Matteo Penegini. Paolo Cascini - Birational geometry and singularities in positive characteristic Many of the results in the Minimal Model Program over complex projective varieties depend on Kodaira's vanishing theorem and its generalizations over singular varieties. Because of the failure of these tools over an algebraically closed field of positive characteristic, it is not known whether these results generalize to this case. The main tool available for the study of varieties over a field of positive characteristic is the Frobenius morphism. In particular, it is crucial to understand singularities from this point of view. I will survey some recent progress in this direction. Cinzia Casagrande - Fano 4-folds, flips, and smooth blow-ups of points Let X be a (smooth, complex) Fano 4-fold. Suppose that there exist a smooth projective 4-fold Y and a point p in Y such that X and the blow-up of Y at p are isomorphic in codimension 1; then the second Betti number of X is at most 12. We will discuss this result and explain the main ideas of the proof. Filippo Favale - A surface of general type which is birational to a surface of degree $10$ of $\mathbb P^3$ I will describe the construction of a family of minimal surfaces of general type with $p_g=q=0$ and $K^2=3$, whose general member $S$ is birational to a surfaces $\Sigma$ of degree $10$ in $\mathbb P^3$. The motivation for studying this type of surfaces comes from some geometric properties that are really interesting. First of all the order of $\pi_1(S)$ is the (conjectured) maximum order for the fundamental group of such surfaces (in this case is $\mathbb Z_4 \ltimes\mathbb Z_4$, and these surfaces are the first discovered with this property). Moreover, the birational map from $S$ to $\Sigma$ is not a morphism and it is given by the linear system $|2K_S+\eta|$ where $\eta$ is a torsion element. Again, this property has never been observed before. Davide Frapporti - On mixed surfaces Let C be a Riemann surface of genus at least 2 and G be a finite group acting on the product $C \times C$ exchanging the factors and such that the index 2 subgroup $G^0$ of the elements that do not exchange the factors acts freely. We call the quotient surface $X:=(C\times C)/G$ a mixed surface. In the talk we investigate these surfaces and explain how their geometry is encoded in the group G. Based on this, we present an algorithm to classify the mixed surfaces with given geometric genus, irregularity, and self-intersection of the canonical class. In particular we give the classification of the mixed surfaces with $K^2>0$ and $p_g=q$ and we discuss their minimality. As a byproduct, we obtain the first examples of minimal surfaces of general type with $K^2=7$ and $p_g=q=1, 2$. This is a joint work with N. Cancian. Andreas Höring - Families of singular rational curves with degenerations A rational curve C on a projective manifold X is said to be minimal if its deformations dominate X and if for a general point in X the deformations of C passing through this point form a proper family. Minimal rational curves have been studied for a long time and have numerous applications to the classification of projective manifolds. In this talk I will introduce a class of rational curves that admits some mild degenerations if we fix two general points. We prove that these curves are immersed in the fixed points which is the analogon of a theorem of Kebekus for minimal rational curves. I will then explain the role of these rational curves for characterizations of hyperquadrics. This is work in progress with Thomas Dedieu. Paolo Lella - Symmetry and equations of the Hilbert scheme The Hilbert scheme is classically realized as a subscheme of a suitable Grassmannian. However, computing the equations of a given Hilbert scheme in terms of Plücker coordinates is a hard achievement. This is mainly due to the dimension of the projective space given by the Plücker embedding and to some redundancy of the Grassmannian. In this talk, I will recall the equations proposed by Iarrobino-Kleiman and Bayer-Haiman-Sturmfels, explaining why it is in fact impossible to compute them explicitly. Then, I will introduce a simpler set of equations (less equations of lower degree) obtained taking into account the symmetries of the Hilbert scheme and I will show some explicit results. This is a joint work with J. Brachat, B. Mourrain and M. Roggero. Massimiliano Mella - On unirational conic bundles A variety is unirational if it is dominated by a rational variety. I aim to investigate the unirationality problem for conic bundles over an arbitrary field. Roberto Muñoz - Weak Zariski decomposition on projective bundles Weak Zariski decomposition on projective bundles The Zariski decomposition of a pseudoeffective divisor on a smooth projective surface is a key ingredient for the study of linear series on surfaces. Since it does not directly apply to higher dimension, several attempts to generalize this notion appear in the literature. One of these generalizations is the weak Zariski decomposition (WZD): a divisor has a WZD if, up to a birational transformation, can be written numerically as the sum of an effective and anef divisor. In this talk we will study this notion and, as a natural application of the description of the nef and pseff cones of certain projective bundles, we will provide some results on the existence of WZD's for divisors on projective bundles over varieties of Picard number one. This is joint work with F. Di Sciullo and L. E. Solá-Conde Matteo Penegini - Characterization of the  5-canonical birationality of algebraic threefolds with $p_g = 3$ In this talk I present a characterization for the birationality of the $5$--canonical map of a minimal algebraic threefolds of general type with $p_g \geq 3$.  Our result is an analogue of the characterization of the $4$-canonical map for surfaces of general type due to Bombieri. This is a joint work with Meng Chen. Sönke Rollenske - Geometry and moduli of stable surfaces The moduli space of stable surfaces is a modular compactification of the Gieseker moduli space of surfaces of general type. I will report joint work with Wenfei Liu, Marco Franciosi and Rita Pardini where we study the geometry of stable surfaces, especially Gorenstein stable surfaces with $K_X^2=1$ Edoardo Sernesi - Syzygies of special line bundles on curves I will introduce geometric conditions called $(\Delta_q)$ on a special very ample line bundle $L$ on a projective curve C. I will show that $(\Delta_3)$ implies that $L$ has the well known property $(M_3)$, generalizing a similar result proved by Voisin for $L=K$. (joint work with M. Aprodu) Luis Solá Conde - Characterizing rational homogeneous spaces among Fano manifolds Many important aspects of the geometry of rational homogeneous spaces, a class including ''classical varieties'' such as projective spaces, quadrics and Grassmannians, can be understood in terms of the representation theory of their groups of automorphisms. On the other hand, in the context of Mori theory, rational homogeneous spaces appear within the broader class of Fano manifolds, hence it is a natural question to characterize them in terms of Mori-theoretical properties, such us the positivity of their tangent bundles, or the behaviour of their famillies of rational curves. In this talk I will make an account on some recent results in this direction, obtained within a joint project with R. Muñoz, G. Occhetta, K. Watanabe and J. Wisniewski. Luca Tasin - Algebraic structures and Chern numbers. In 1952, F. Hirzebruch posed the question about the topological invariance of Chern numbers of complex projective varieties. D. Kotschick in 2012 solved the problem and  asked the following question: which Chern numbers are determined up to finite ambiguity by the underlying smooth manifold? We will show that in dimension higher than 3 only few Chern numbers are bounded by the underlying manifold. Then we will analyse the 3-dimensional case, where the minimal model program plays a major role in our approach to this problem. Filippo Viviani - Fourier-Mukai and autoduality for compactified Jacobians To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) singular Calabi-Yau projective varieties, each of which yields a modular compactification of a disjoint union of copies of the generalized Jacobian of X. We define a Poincaré sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the associated integral transform is an equivalence of their derived categories, hence it defines a Fourier-Mukai transform. This generalizes the classical result of S. Mukai for Jacobians of smooth curves and the more recent result of D. Arinkin for compactified Jacobians of integral curves with planar singularities, and it provides further evidence  for the classical limit of the geometric Langlands conjecture (as formulated by Donagi and Pantev). As a corollary, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that  algebraic equivalence and numerical equivalence coincide on any fine compactified Jacobian. This is a joint work with M. Melo and A. Rapagnetta.