We introduce a method to produce bounds for the non secant defectivity of an arbitrary irreducible projective variety, once we know how its osculating spaces behave in families and when the linear projections from them are generically finite.Then we analyze the relative dimension of osculating projections of Grassmannians, and as an application of our techniques we prove that asymptotically the Grassmannian G(r, n), parametrizing r-planes in P n , is not h-defective for h ≤ ( n+1 r+1 ) ⌊log 2 (r)⌋ . This bound improves the previous one h ≤ n−r 3 + 1, due to H. Abo, G. Ottaviani and C. Peterson, for any r ≥ 4.Contents 6 3. Osculating projections 9 4. Degenerating tangential projections to osculating projections 13 5. Non secant defectivity via osculating projections 24 References 28

Let SV n n n d d d be the Segre-Veronese given as the image of the embedding induced by the line bundle O P n 1 ×···×P nr (d1, . . . , dr). We prove that asymptotically SV n n n d d d is not h-defective for h ≤ n ⌊log 2 (d−1)⌋

The distributive sequential n-site phosphorylation/dephosphorylation system is an important building block in networks of chemical reactions arising in molecular biology, which has been intensively studied. In the nice paper of Wang and Sontag (2008) it is shown that for certain choices of the reaction rate constants and total conservation constants, the system can have 2[ n 2 ] + 1 positive steady states (that is, n + 1 positive steady states for n even and n positive steady states for n odd). In this paper we give open parameter regions in the space of reaction rate constants and total conservation constants that ensure these number of positive steady states, while assuming in the modeling that roughly only 1 4 of the intermediates occur in the reaction mechanism. This result is based on the general framework developed by Bihan, Dickenstein, and Giaroli (2018), which can be applied to other networks. We also describe how to implement these tools to search for multistationarity regions in a computer algebra system and present some computer aided results.

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The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the geometrical and the combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our criterion is sharp. Finally, we classify the smooth toric 3-folds of Picard rank three for which the two decompositions are different.