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SUMMARY:Kaloyan Slavov (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201021T183000Z
DTEND;VALUE=DATE-TIME:20201021T193000Z
DTSTAMP;VALUE=DATE-TIME:20240329T101717Z
UID:Lecture_series_NT_AG/1
DESCRIPTION:Title: An application of random plane slicing to counting $\\mathbb{
F}_q$-points on hypersurfaces\nby Kaloyan Slavov (ETH Zurich) as part
of Lecture series in number theory and algebraic geometry\n\n\nAbstract\nW
e first review the classical Lang--Weil bound on the number of $\\mathbb{F
}_q$-points on a geometrically irreducible hypersurface $X$ over a finite
field $\\mathbb{F}_q$. By studying the intersection of $X(\\mathbb{F}_q)$
with a random $\\mathbb{F}_q$-plane\, we improve the best known bounds in
the literature for $|X(\\mathbb{F}_q)|$.\n
LOCATION:https://researchseminars.org/talk/Lecture_series_NT_AG/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaloyan Slavov (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201026T200000Z
DTEND;VALUE=DATE-TIME:20201026T210000Z
DTSTAMP;VALUE=DATE-TIME:20240329T101717Z
UID:Lecture_series_NT_AG/2
DESCRIPTION:Title: A refinement of Bertini irreducibility via point-counting ove
r finite fields\nby Kaloyan Slavov (ETH Zurich) as part of Lecture ser
ies in number theory and algebraic geometry\n\n\nAbstract\nWe approach cla
ssical Bertini irreducibility theorems over an arbitrary algebraically clo
sed field through a reduction to point-counting over finite fields and a p
robabilistic combinatorial argument based on random hyperplane slicing. A
classical theorem by Bertini states that if $X\\subset\\mathbb{P}^n$ is an
irreducible variety of dimension at least $2$\, then there is a dense ope
n subset $M_{\\text{good}}$ inside the space \n$\\check{\\mathbb{P}}^n$ of
hyperplanes in $\\mathbb{P}^n$ such that $X\\cap H$ is irreducible for ea
ch $H$ in $M_{\\text{good}}$. Benoist proved that in fact\, the complement
of $M_{\\text{good}}$ in \n$\\check{\\mathbb{P}}^n$ has dimension at most
$\\operatorname{codim} X+1$. We give a new proof of this\, along with a r
efinement in which the embedding $X\\hookrightarrow\\mathbb{P}^n$ is repla
ced by a more general morphism $X\\to\\mathbb{P}^n$. This is joint work wi
th Bjorn Poonen.\n
LOCATION:https://researchseminars.org/talk/Lecture_series_NT_AG/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaloyan Slavov (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201028T183000Z
DTEND;VALUE=DATE-TIME:20201028T193000Z
DTSTAMP;VALUE=DATE-TIME:20240329T101717Z
UID:Lecture_series_NT_AG/3
DESCRIPTION:Title: The moduli space of hypersurfaces whose singular locus has hi
gh dimension\nby Kaloyan Slavov (ETH Zurich) as part of Lecture series
in number theory and algebraic geometry\n\n\nAbstract\nConsider the modul
i space of hypersurfaces of degree $\\ell$ in $\\mathbb{P}^n$ whose singul
ar locus has dimension at least $b$ (for a fixed $b\\geq 1$). We prove tha
t when $\\ell$ is large\, this moduli space has a unique irreducible compo
nent of maximal dimension\, consisting of the hypersurfaces singular along
a linear $b$-​dimensional space. The proof will involve a reduction to
positive characteristic and a probabilistic counting argument over finite
fields.\n
LOCATION:https://researchseminars.org/talk/Lecture_series_NT_AG/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaloyan Slavov (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201102T210000Z
DTEND;VALUE=DATE-TIME:20201102T220000Z
DTSTAMP;VALUE=DATE-TIME:20240329T101717Z
UID:Lecture_series_NT_AG/4
DESCRIPTION:Title: What is the probability that a random (sparse) polynomial of
degree $d$ over a finite field is irreducible?\nby Kaloyan Slavov (ETH
Zurich) as part of Lecture series in number theory and algebraic geometry
\n\n\nAbstract\nA classical result of Gauss states that among all monic po
lynomials of degree $d$ over a finite field\,\napproximately $1/d$ are irr
educible. Extending previous results in the literature\, we prove that und
er a mild assumption\, the proportion of irreducible polynomials does not
change even if only the last two coefficients are allowed to vary. Our app
roach is geometric. The talk will be nontechnical and accessible to a broa
d audience.\n
LOCATION:https://researchseminars.org/talk/Lecture_series_NT_AG/4/
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